Add files using upload-large-folder tool

llmeval-env/lib/python3.10/site-packages/sympy/matrices/__init__.py

@@ -0,0 +1,71 @@
+"""A module that handles matrices.
+
+Includes functions for fast creating matrices like zero, one/eye, random
+matrix, etc.
+"""
+from .common import ShapeError, NonSquareMatrixError, MatrixKind
+from .dense import (
+    GramSchmidt, casoratian, diag, eye, hessian, jordan_cell,
+    list2numpy, matrix2numpy, matrix_multiply_elementwise, ones,
+    randMatrix, rot_axis1, rot_axis2, rot_axis3, rot_ccw_axis1,
+    rot_ccw_axis2, rot_ccw_axis3, rot_givens,
+    symarray, wronskian, zeros)
+from .dense import MutableDenseMatrix
+from .matrices import DeferredVector, MatrixBase
+
+MutableMatrix = MutableDenseMatrix
+Matrix = MutableMatrix
+
+from .sparse import MutableSparseMatrix
+from .sparsetools import banded
+from .immutable import ImmutableDenseMatrix, ImmutableSparseMatrix
+
+ImmutableMatrix = ImmutableDenseMatrix
+SparseMatrix = MutableSparseMatrix
+
+from .expressions import (
+    MatrixSlice, BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity,
+    Inverse, MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace,
+    Transpose, ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols, Adjoint,
+    hadamard_product, HadamardProduct, HadamardPower, Determinant, det,
+    diagonalize_vector, DiagMatrix, DiagonalMatrix, DiagonalOf, trace,
+    DotProduct, kronecker_product, KroneckerProduct,
+    PermutationMatrix, MatrixPermute, MatrixSet, Permanent, per)
+
+from .utilities import dotprodsimp
+
+__all__ = [
+    'ShapeError', 'NonSquareMatrixError', 'MatrixKind',
+
+    'GramSchmidt', 'casoratian', 'diag', 'eye', 'hessian', 'jordan_cell',
+    'list2numpy', 'matrix2numpy', 'matrix_multiply_elementwise', 'ones',
+    'randMatrix', 'rot_axis1', 'rot_axis2', 'rot_axis3', 'symarray',
+    'wronskian', 'zeros', 'rot_ccw_axis1', 'rot_ccw_axis2', 'rot_ccw_axis3',
+    'rot_givens',
+
+    'MutableDenseMatrix',
+
+    'DeferredVector', 'MatrixBase',
+
+    'Matrix', 'MutableMatrix',
+
+    'MutableSparseMatrix',
+
+    'banded',
+
+    'ImmutableDenseMatrix', 'ImmutableSparseMatrix',
+
+    'ImmutableMatrix', 'SparseMatrix',
+
+    'MatrixSlice', 'BlockDiagMatrix', 'BlockMatrix', 'FunctionMatrix',
+    'Identity', 'Inverse', 'MatAdd', 'MatMul', 'MatPow', 'MatrixExpr',
+    'MatrixSymbol', 'Trace', 'Transpose', 'ZeroMatrix', 'OneMatrix',
+    'blockcut', 'block_collapse', 'matrix_symbols', 'Adjoint',
+    'hadamard_product', 'HadamardProduct', 'HadamardPower', 'Determinant',
+    'det', 'diagonalize_vector', 'DiagMatrix', 'DiagonalMatrix',
+    'DiagonalOf', 'trace', 'DotProduct', 'kronecker_product',
+    'KroneckerProduct', 'PermutationMatrix', 'MatrixPermute', 'MatrixSet',
+    'Permanent', 'per',
+
+    'dotprodsimp',
+]

llmeval-env/lib/python3.10/site-packages/sympy/matrices/benchmarks/bench_matrix.py

@@ -0,0 +1,21 @@
+from sympy.core.numbers import Integer
+from sympy.matrices.dense import (eye, zeros)
+
+i3 = Integer(3)
+M = eye(100)
+
+
+def timeit_Matrix__getitem_ii():
+    M[3, 3]
+
+
+def timeit_Matrix__getitem_II():
+    M[i3, i3]
+
+
+def timeit_Matrix__getslice():
+    M[:, :]
+
+
+def timeit_Matrix_zeronm():
+    zeros(100, 100)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/common.py

@@ -0,0 +1,3227 @@
+"""
+Basic methods common to all matrices to be used
+when creating more advanced matrices (e.g., matrices over rings,
+etc.).
+"""
+
+from collections import defaultdict
+from collections.abc import Iterable
+from inspect import isfunction
+from functools import reduce
+
+from sympy.assumptions.refine import refine
+from sympy.core import SympifyError, Add
+from sympy.core.basic import Atom
+from sympy.core.decorators import call_highest_priority
+from sympy.core.kind import Kind, NumberKind
+from sympy.core.logic import fuzzy_and, FuzzyBool
+from sympy.core.mod import Mod
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.core.sympify import sympify
+from sympy.functions.elementary.complexes import Abs, re, im
+from .utilities import _dotprodsimp, _simplify
+from sympy.polys.polytools import Poly
+from sympy.utilities.iterables import flatten, is_sequence
+from sympy.utilities.misc import as_int, filldedent
+from sympy.tensor.array import NDimArray
+
+from .utilities import _get_intermediate_simp_bool
+
+
+class MatrixError(Exception):
+    pass
+
+
+class ShapeError(ValueError, MatrixError):
+    """Wrong matrix shape"""
+    pass
+
+
+class NonSquareMatrixError(ShapeError):
+    pass
+
+
+class NonInvertibleMatrixError(ValueError, MatrixError):
+    """The matrix in not invertible (division by multidimensional zero error)."""
+    pass
+
+
+class NonPositiveDefiniteMatrixError(ValueError, MatrixError):
+    """The matrix is not a positive-definite matrix."""
+    pass
+
+
+class MatrixRequired:
+    """All subclasses of matrix objects must implement the
+    required matrix properties listed here."""
+    rows = None  # type: int
+    cols = None  # type: int
+    _simplify = None
+
+    @classmethod
+    def _new(cls, *args, **kwargs):
+        """`_new` must, at minimum, be callable as
+        `_new(rows, cols, mat) where mat is a flat list of the
+        elements of the matrix."""
+        raise NotImplementedError("Subclasses must implement this.")
+
+    def __eq__(self, other):
+        raise NotImplementedError("Subclasses must implement this.")
+
+    def __getitem__(self, key):
+        """Implementations of __getitem__ should accept ints, in which
+        case the matrix is indexed as a flat list, tuples (i,j) in which
+        case the (i,j) entry is returned, slices, or mixed tuples (a,b)
+        where a and b are any combination of slices and integers."""
+        raise NotImplementedError("Subclasses must implement this.")
+
+    def __len__(self):
+        """The total number of entries in the matrix."""
+        raise NotImplementedError("Subclasses must implement this.")
+
+    @property
+    def shape(self):
+        raise NotImplementedError("Subclasses must implement this.")
+
+
+class MatrixShaping(MatrixRequired):
+    """Provides basic matrix shaping and extracting of submatrices"""
+
+    def _eval_col_del(self, col):
+        def entry(i, j):
+            return self[i, j] if j < col else self[i, j + 1]
+        return self._new(self.rows, self.cols - 1, entry)
+
+    def _eval_col_insert(self, pos, other):
+
+        def entry(i, j):
+            if j < pos:
+                return self[i, j]
+            elif pos <= j < pos + other.cols:
+                return other[i, j - pos]
+            return self[i, j - other.cols]
+
+        return self._new(self.rows, self.cols + other.cols, entry)
+
+    def _eval_col_join(self, other):
+        rows = self.rows
+
+        def entry(i, j):
+            if i < rows:
+                return self[i, j]
+            return other[i - rows, j]
+
+        return classof(self, other)._new(self.rows + other.rows, self.cols,
+                                         entry)
+
+    def _eval_extract(self, rowsList, colsList):
+        mat = list(self)
+        cols = self.cols
+        indices = (i * cols + j for i in rowsList for j in colsList)
+        return self._new(len(rowsList), len(colsList),
+                         [mat[i] for i in indices])
+
+    def _eval_get_diag_blocks(self):
+        sub_blocks = []
+
+        def recurse_sub_blocks(M):
+            i = 1
+            while i <= M.shape[0]:
+                if i == 1:
+                    to_the_right = M[0, i:]
+                    to_the_bottom = M[i:, 0]
+                else:
+                    to_the_right = M[:i, i:]
+                    to_the_bottom = M[i:, :i]
+                if any(to_the_right) or any(to_the_bottom):
+                    i += 1
+                    continue
+                else:
+                    sub_blocks.append(M[:i, :i])
+                    if M.shape == M[:i, :i].shape:
+                        return
+                    else:
+                        recurse_sub_blocks(M[i:, i:])
+                        return
+
+        recurse_sub_blocks(self)
+        return sub_blocks
+
+    def _eval_row_del(self, row):
+        def entry(i, j):
+            return self[i, j] if i < row else self[i + 1, j]
+        return self._new(self.rows - 1, self.cols, entry)
+
+    def _eval_row_insert(self, pos, other):
+        entries = list(self)
+        insert_pos = pos * self.cols
+        entries[insert_pos:insert_pos] = list(other)
+        return self._new(self.rows + other.rows, self.cols, entries)
+
+    def _eval_row_join(self, other):
+        cols = self.cols
+
+        def entry(i, j):
+            if j < cols:
+                return self[i, j]
+            return other[i, j - cols]
+
+        return classof(self, other)._new(self.rows, self.cols + other.cols,
+                                         entry)
+
+    def _eval_tolist(self):
+        return [list(self[i,:]) for i in range(self.rows)]
+
+    def _eval_todok(self):
+        dok = {}
+        rows, cols = self.shape
+        for i in range(rows):
+            for j in range(cols):
+                val = self[i, j]
+                if val != self.zero:
+                    dok[i, j] = val
+        return dok
+
+    def _eval_vec(self):
+        rows = self.rows
+
+        def entry(n, _):
+            # we want to read off the columns first
+            j = n // rows
+            i = n - j * rows
+            return self[i, j]
+
+        return self._new(len(self), 1, entry)
+
+    def _eval_vech(self, diagonal):
+        c = self.cols
+        v = []
+        if diagonal:
+            for j in range(c):
+                for i in range(j, c):
+                    v.append(self[i, j])
+        else:
+            for j in range(c):
+                for i in range(j + 1, c):
+                    v.append(self[i, j])
+        return self._new(len(v), 1, v)
+
+    def col_del(self, col):
+        """Delete the specified column."""
+        if col < 0:
+            col += self.cols
+        if not 0 <= col < self.cols:
+            raise IndexError("Column {} is out of range.".format(col))
+        return self._eval_col_del(col)
+
+    def col_insert(self, pos, other):
+        """Insert one or more columns at the given column position.
+
+        Examples
+        ========
+
+        >>> from sympy import zeros, ones
+        >>> M = zeros(3)
+        >>> V = ones(3, 1)
+        >>> M.col_insert(1, V)
+        Matrix([
+        [0, 1, 0, 0],
+        [0, 1, 0, 0],
+        [0, 1, 0, 0]])
+
+        See Also
+        ========
+
+        col
+        row_insert
+        """
+        # Allows you to build a matrix even if it is null matrix
+        if not self:
+            return type(self)(other)
+
+        pos = as_int(pos)
+
+        if pos < 0:
+            pos = self.cols + pos
+        if pos < 0:
+            pos = 0
+        elif pos > self.cols:
+            pos = self.cols
+
+        if self.rows != other.rows:
+            raise ShapeError(
+                "The matrices have incompatible number of rows ({} and {})"
+                .format(self.rows, other.rows))
+
+        return self._eval_col_insert(pos, other)
+
+    def col_join(self, other):
+        """Concatenates two matrices along self's last and other's first row.
+
+        Examples
+        ========
+
+        >>> from sympy import zeros, ones
+        >>> M = zeros(3)
+        >>> V = ones(1, 3)
+        >>> M.col_join(V)
+        Matrix([
+        [0, 0, 0],
+        [0, 0, 0],
+        [0, 0, 0],
+        [1, 1, 1]])
+
+        See Also
+        ========
+
+        col
+        row_join
+        """
+        # A null matrix can always be stacked (see  #10770)
+        if self.rows == 0 and self.cols != other.cols:
+            return self._new(0, other.cols, []).col_join(other)
+
+        if self.cols != other.cols:
+            raise ShapeError(
+                "The matrices have incompatible number of columns ({} and {})"
+                .format(self.cols, other.cols))
+        return self._eval_col_join(other)
+
+    def col(self, j):
+        """Elementary column selector.
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> eye(2).col(0)
+        Matrix([
+        [1],
+        [0]])
+
+        See Also
+        ========
+
+        row
+        col_del
+        col_join
+        col_insert
+        """
+        return self[:, j]
+
+    def extract(self, rowsList, colsList):
+        r"""Return a submatrix by specifying a list of rows and columns.
+        Negative indices can be given. All indices must be in the range
+        $-n \le i < n$ where $n$ is the number of rows or columns.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(4, 3, range(12))
+        >>> m
+        Matrix([
+        [0,  1,  2],
+        [3,  4,  5],
+        [6,  7,  8],
+        [9, 10, 11]])
+        >>> m.extract([0, 1, 3], [0, 1])
+        Matrix([
+        [0,  1],
+        [3,  4],
+        [9, 10]])
+
+        Rows or columns can be repeated:
+
+        >>> m.extract([0, 0, 1], [-1])
+        Matrix([
+        [2],
+        [2],
+        [5]])
+
+        Every other row can be taken by using range to provide the indices:
+
+        >>> m.extract(range(0, m.rows, 2), [-1])
+        Matrix([
+        [2],
+        [8]])
+
+        RowsList or colsList can also be a list of booleans, in which case
+        the rows or columns corresponding to the True values will be selected:
+
+        >>> m.extract([0, 1, 2, 3], [True, False, True])
+        Matrix([
+        [0,  2],
+        [3,  5],
+        [6,  8],
+        [9, 11]])
+        """
+
+        if not is_sequence(rowsList) or not is_sequence(colsList):
+            raise TypeError("rowsList and colsList must be iterable")
+        # ensure rowsList and colsList are lists of integers
+        if rowsList and all(isinstance(i, bool) for i in rowsList):
+            rowsList = [index for index, item in enumerate(rowsList) if item]
+        if colsList and all(isinstance(i, bool) for i in colsList):
+            colsList = [index for index, item in enumerate(colsList) if item]
+
+        # ensure everything is in range
+        rowsList = [a2idx(k, self.rows) for k in rowsList]
+        colsList = [a2idx(k, self.cols) for k in colsList]
+
+        return self._eval_extract(rowsList, colsList)
+
+    def get_diag_blocks(self):
+        """Obtains the square sub-matrices on the main diagonal of a square matrix.
+
+        Useful for inverting symbolic matrices or solving systems of
+        linear equations which may be decoupled by having a block diagonal
+        structure.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> from sympy.abc import x, y, z
+        >>> A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]])
+        >>> a1, a2, a3 = A.get_diag_blocks()
+        >>> a1
+        Matrix([
+        [1,    3],
+        [y, z**2]])
+        >>> a2
+        Matrix([[x]])
+        >>> a3
+        Matrix([[0]])
+
+        """
+        return self._eval_get_diag_blocks()
+
+    @classmethod
+    def hstack(cls, *args):
+        """Return a matrix formed by joining args horizontally (i.e.
+        by repeated application of row_join).
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, eye
+        >>> Matrix.hstack(eye(2), 2*eye(2))
+        Matrix([
+        [1, 0, 2, 0],
+        [0, 1, 0, 2]])
+        """
+        if len(args) == 0:
+            return cls._new()
+
+        kls = type(args[0])
+        return reduce(kls.row_join, args)
+
+    def reshape(self, rows, cols):
+        """Reshape the matrix. Total number of elements must remain the same.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(2, 3, lambda i, j: 1)
+        >>> m
+        Matrix([
+        [1, 1, 1],
+        [1, 1, 1]])
+        >>> m.reshape(1, 6)
+        Matrix([[1, 1, 1, 1, 1, 1]])
+        >>> m.reshape(3, 2)
+        Matrix([
+        [1, 1],
+        [1, 1],
+        [1, 1]])
+
+        """
+        if self.rows * self.cols != rows * cols:
+            raise ValueError("Invalid reshape parameters %d %d" % (rows, cols))
+        return self._new(rows, cols, lambda i, j: self[i * cols + j])
+
+    def row_del(self, row):
+        """Delete the specified row."""
+        if row < 0:
+            row += self.rows
+        if not 0 <= row < self.rows:
+            raise IndexError("Row {} is out of range.".format(row))
+
+        return self._eval_row_del(row)
+
+    def row_insert(self, pos, other):
+        """Insert one or more rows at the given row position.
+
+        Examples
+        ========
+
+        >>> from sympy import zeros, ones
+        >>> M = zeros(3)
+        >>> V = ones(1, 3)
+        >>> M.row_insert(1, V)
+        Matrix([
+        [0, 0, 0],
+        [1, 1, 1],
+        [0, 0, 0],
+        [0, 0, 0]])
+
+        See Also
+        ========
+
+        row
+        col_insert
+        """
+        # Allows you to build a matrix even if it is null matrix
+        if not self:
+            return self._new(other)
+
+        pos = as_int(pos)
+
+        if pos < 0:
+            pos = self.rows + pos
+        if pos < 0:
+            pos = 0
+        elif pos > self.rows:
+            pos = self.rows
+
+        if self.cols != other.cols:
+            raise ShapeError(
+                "The matrices have incompatible number of columns ({} and {})"
+                .format(self.cols, other.cols))
+
+        return self._eval_row_insert(pos, other)
+
+    def row_join(self, other):
+        """Concatenates two matrices along self's last and rhs's first column
+
+        Examples
+        ========
+
+        >>> from sympy import zeros, ones
+        >>> M = zeros(3)
+        >>> V = ones(3, 1)
+        >>> M.row_join(V)
+        Matrix([
+        [0, 0, 0, 1],
+        [0, 0, 0, 1],
+        [0, 0, 0, 1]])
+
+        See Also
+        ========
+
+        row
+        col_join
+        """
+        # A null matrix can always be stacked (see  #10770)
+        if self.cols == 0 and self.rows != other.rows:
+            return self._new(other.rows, 0, []).row_join(other)
+
+        if self.rows != other.rows:
+            raise ShapeError(
+                "The matrices have incompatible number of rows ({} and {})"
+                .format(self.rows, other.rows))
+        return self._eval_row_join(other)
+
+    def diagonal(self, k=0):
+        """Returns the kth diagonal of self. The main diagonal
+        corresponds to `k=0`; diagonals above and below correspond to
+        `k > 0` and `k < 0`, respectively. The values of `self[i, j]`
+        for which `j - i = k`, are returned in order of increasing
+        `i + j`, starting with `i + j = |k|`.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(3, 3, lambda i, j: j - i); m
+        Matrix([
+        [ 0,  1, 2],
+        [-1,  0, 1],
+        [-2, -1, 0]])
+        >>> _.diagonal()
+        Matrix([[0, 0, 0]])
+        >>> m.diagonal(1)
+        Matrix([[1, 1]])
+        >>> m.diagonal(-2)
+        Matrix([[-2]])
+
+        Even though the diagonal is returned as a Matrix, the element
+        retrieval can be done with a single index:
+
+        >>> Matrix.diag(1, 2, 3).diagonal()[1]  # instead of [0, 1]
+        2
+
+        See Also
+        ========
+
+        diag
+        """
+        rv = []
+        k = as_int(k)
+        r = 0 if k > 0 else -k
+        c = 0 if r else k
+        while True:
+            if r == self.rows or c == self.cols:
+                break
+            rv.append(self[r, c])
+            r += 1
+            c += 1
+        if not rv:
+            raise ValueError(filldedent('''
+            The %s diagonal is out of range [%s, %s]''' % (
+            k, 1 - self.rows, self.cols - 1)))
+        return self._new(1, len(rv), rv)
+
+    def row(self, i):
+        """Elementary row selector.
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> eye(2).row(0)
+        Matrix([[1, 0]])
+
+        See Also
+        ========
+
+        col
+        row_del
+        row_join
+        row_insert
+        """
+        return self[i, :]
+
+    @property
+    def shape(self):
+        """The shape (dimensions) of the matrix as the 2-tuple (rows, cols).
+
+        Examples
+        ========
+
+        >>> from sympy import zeros
+        >>> M = zeros(2, 3)
+        >>> M.shape
+        (2, 3)
+        >>> M.rows
+        2
+        >>> M.cols
+        3
+        """
+        return (self.rows, self.cols)
+
+    def todok(self):
+        """Return the matrix as dictionary of keys.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> M = Matrix.eye(3)
+        >>> M.todok()
+        {(0, 0): 1, (1, 1): 1, (2, 2): 1}
+        """
+        return self._eval_todok()
+
+    def tolist(self):
+        """Return the Matrix as a nested Python list.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, ones
+        >>> m = Matrix(3, 3, range(9))
+        >>> m
+        Matrix([
+        [0, 1, 2],
+        [3, 4, 5],
+        [6, 7, 8]])
+        >>> m.tolist()
+        [[0, 1, 2], [3, 4, 5], [6, 7, 8]]
+        >>> ones(3, 0).tolist()
+        [[], [], []]
+
+        When there are no rows then it will not be possible to tell how
+        many columns were in the original matrix:
+
+        >>> ones(0, 3).tolist()
+        []
+
+        """
+        if not self.rows:
+            return []
+        if not self.cols:
+            return [[] for i in range(self.rows)]
+        return self._eval_tolist()
+
+    def todod(M):
+        """Returns matrix as dict of dicts containing non-zero elements of the Matrix
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix([[0, 1],[0, 3]])
+        >>> A
+        Matrix([
+        [0, 1],
+        [0, 3]])
+        >>> A.todod()
+        {0: {1: 1}, 1: {1: 3}}
+
+
+        """
+        rowsdict = {}
+        Mlol = M.tolist()
+        for i, Mi in enumerate(Mlol):
+            row = {j: Mij for j, Mij in enumerate(Mi) if Mij}
+            if row:
+                rowsdict[i] = row
+        return rowsdict
+
+    def vec(self):
+        """Return the Matrix converted into a one column matrix by stacking columns
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m=Matrix([[1, 3], [2, 4]])
+        >>> m
+        Matrix([
+        [1, 3],
+        [2, 4]])
+        >>> m.vec()
+        Matrix([
+        [1],
+        [2],
+        [3],
+        [4]])
+
+        See Also
+        ========
+
+        vech
+        """
+        return self._eval_vec()
+
+    def vech(self, diagonal=True, check_symmetry=True):
+        """Reshapes the matrix into a column vector by stacking the
+        elements in the lower triangle.
+
+        Parameters
+        ==========
+
+        diagonal : bool, optional
+            If ``True``, it includes the diagonal elements.
+
+        check_symmetry : bool, optional
+            If ``True``, it checks whether the matrix is symmetric.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m=Matrix([[1, 2], [2, 3]])
+        >>> m
+        Matrix([
+        [1, 2],
+        [2, 3]])
+        >>> m.vech()
+        Matrix([
+        [1],
+        [2],
+        [3]])
+        >>> m.vech(diagonal=False)
+        Matrix([[2]])
+
+        Notes
+        =====
+
+        This should work for symmetric matrices and ``vech`` can
+        represent symmetric matrices in vector form with less size than
+        ``vec``.
+
+        See Also
+        ========
+
+        vec
+        """
+        if not self.is_square:
+            raise NonSquareMatrixError
+
+        if check_symmetry and not self.is_symmetric():
+            raise ValueError("The matrix is not symmetric.")
+
+        return self._eval_vech(diagonal)
+
+    @classmethod
+    def vstack(cls, *args):
+        """Return a matrix formed by joining args vertically (i.e.
+        by repeated application of col_join).
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, eye
+        >>> Matrix.vstack(eye(2), 2*eye(2))
+        Matrix([
+        [1, 0],
+        [0, 1],
+        [2, 0],
+        [0, 2]])
+        """
+        if len(args) == 0:
+            return cls._new()
+
+        kls = type(args[0])
+        return reduce(kls.col_join, args)
+
+
+class MatrixSpecial(MatrixRequired):
+    """Construction of special matrices"""
+
+    @classmethod
+    def _eval_diag(cls, rows, cols, diag_dict):
+        """diag_dict is a defaultdict containing
+        all the entries of the diagonal matrix."""
+        def entry(i, j):
+            return diag_dict[(i, j)]
+        return cls._new(rows, cols, entry)
+
+    @classmethod
+    def _eval_eye(cls, rows, cols):
+        vals = [cls.zero]*(rows*cols)
+        vals[::cols+1] = [cls.one]*min(rows, cols)
+        return cls._new(rows, cols, vals, copy=False)
+
+    @classmethod
+    def _eval_jordan_block(cls, size: int, eigenvalue, band='upper'):
+        if band == 'lower':
+            def entry(i, j):
+                if i == j:
+                    return eigenvalue
+                elif j + 1 == i:
+                    return cls.one
+                return cls.zero
+        else:
+            def entry(i, j):
+                if i == j:
+                    return eigenvalue
+                elif i + 1 == j:
+                    return cls.one
+                return cls.zero
+        return cls._new(size, size, entry)
+
+    @classmethod
+    def _eval_ones(cls, rows, cols):
+        def entry(i, j):
+            return cls.one
+        return cls._new(rows, cols, entry)
+
+    @classmethod
+    def _eval_zeros(cls, rows, cols):
+        return cls._new(rows, cols, [cls.zero]*(rows*cols), copy=False)
+
+    @classmethod
+    def _eval_wilkinson(cls, n):
+        def entry(i, j):
+            return cls.one if i + 1 == j else cls.zero
+
+        D = cls._new(2*n + 1, 2*n + 1, entry)
+
+        wminus = cls.diag(list(range(-n, n + 1)), unpack=True) + D + D.T
+        wplus = abs(cls.diag(list(range(-n, n + 1)), unpack=True)) + D + D.T
+
+        return wminus, wplus
+
+    @classmethod
+    def diag(kls, *args, strict=False, unpack=True, rows=None, cols=None, **kwargs):
+        """Returns a matrix with the specified diagonal.
+        If matrices are passed, a block-diagonal matrix
+        is created (i.e. the "direct sum" of the matrices).
+
+        kwargs
+        ======
+
+        rows : rows of the resulting matrix; computed if
+               not given.
+
+        cols : columns of the resulting matrix; computed if
+               not given.
+
+        cls : class for the resulting matrix
+
+        unpack : bool which, when True (default), unpacks a single
+        sequence rather than interpreting it as a Matrix.
+
+        strict : bool which, when False (default), allows Matrices to
+        have variable-length rows.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> Matrix.diag(1, 2, 3)
+        Matrix([
+        [1, 0, 0],
+        [0, 2, 0],
+        [0, 0, 3]])
+
+        The current default is to unpack a single sequence. If this is
+        not desired, set `unpack=False` and it will be interpreted as
+        a matrix.
+
+        >>> Matrix.diag([1, 2, 3]) == Matrix.diag(1, 2, 3)
+        True
+
+        When more than one element is passed, each is interpreted as
+        something to put on the diagonal. Lists are converted to
+        matrices. Filling of the diagonal always continues from
+        the bottom right hand corner of the previous item: this
+        will create a block-diagonal matrix whether the matrices
+        are square or not.
+
+        >>> col = [1, 2, 3]
+        >>> row = [[4, 5]]
+        >>> Matrix.diag(col, row)
+        Matrix([
+        [1, 0, 0],
+        [2, 0, 0],
+        [3, 0, 0],
+        [0, 4, 5]])
+
+        When `unpack` is False, elements within a list need not all be
+        of the same length. Setting `strict` to True would raise a
+        ValueError for the following:
+
+        >>> Matrix.diag([[1, 2, 3], [4, 5], [6]], unpack=False)
+        Matrix([
+        [1, 2, 3],
+        [4, 5, 0],
+        [6, 0, 0]])
+
+        The type of the returned matrix can be set with the ``cls``
+        keyword.
+
+        >>> from sympy import ImmutableMatrix
+        >>> from sympy.utilities.misc import func_name
+        >>> func_name(Matrix.diag(1, cls=ImmutableMatrix))
+        'ImmutableDenseMatrix'
+
+        A zero dimension matrix can be used to position the start of
+        the filling at the start of an arbitrary row or column:
+
+        >>> from sympy import ones
+        >>> r2 = ones(0, 2)
+        >>> Matrix.diag(r2, 1, 2)
+        Matrix([
+        [0, 0, 1, 0],
+        [0, 0, 0, 2]])
+
+        See Also
+        ========
+        eye
+        diagonal
+        .dense.diag
+        .expressions.blockmatrix.BlockMatrix
+        .sparsetools.banded
+       """
+        from sympy.matrices.matrices import MatrixBase
+        from sympy.matrices.dense import Matrix
+        from sympy.matrices import SparseMatrix
+        klass = kwargs.get('cls', kls)
+        if unpack and len(args) == 1 and is_sequence(args[0]) and \
+                not isinstance(args[0], MatrixBase):
+            args = args[0]
+
+        # fill a default dict with the diagonal entries
+        diag_entries = defaultdict(int)
+        rmax = cmax = 0  # keep track of the biggest index seen
+        for m in args:
+            if isinstance(m, list):
+                if strict:
+                    # if malformed, Matrix will raise an error
+                    _ = Matrix(m)
+                    r, c = _.shape
+                    m = _.tolist()
+                else:
+                    r, c, smat = SparseMatrix._handle_creation_inputs(m)
+                    for (i, j), _ in smat.items():
+                        diag_entries[(i + rmax, j + cmax)] = _
+                    m = []  # to skip process below
+            elif hasattr(m, 'shape'):  # a Matrix
+                # convert to list of lists
+                r, c = m.shape
+                m = m.tolist()
+            else:  # in this case, we're a single value
+                diag_entries[(rmax, cmax)] = m
+                rmax += 1
+                cmax += 1
+                continue
+            # process list of lists
+            for i, mi in enumerate(m):
+                for j, _ in enumerate(mi):
+                    diag_entries[(i + rmax, j + cmax)] = _
+            rmax += r
+            cmax += c
+        if rows is None:
+            rows, cols = cols, rows
+        if rows is None:
+            rows, cols = rmax, cmax
+        else:
+            cols = rows if cols is None else cols
+        if rows < rmax or cols < cmax:
+            raise ValueError(filldedent('''
+                The constructed matrix is {} x {} but a size of {} x {}
+                was specified.'''.format(rmax, cmax, rows, cols)))
+        return klass._eval_diag(rows, cols, diag_entries)
+
+    @classmethod
+    def eye(kls, rows, cols=None, **kwargs):
+        """Returns an identity matrix.
+
+        Parameters
+        ==========
+
+        rows : rows of the matrix
+        cols : cols of the matrix (if None, cols=rows)
+
+        kwargs
+        ======
+        cls : class of the returned matrix
+        """
+        if cols is None:
+            cols = rows
+        if rows < 0 or cols < 0:
+            raise ValueError("Cannot create a {} x {} matrix. "
+                             "Both dimensions must be positive".format(rows, cols))
+        klass = kwargs.get('cls', kls)
+        rows, cols = as_int(rows), as_int(cols)
+
+        return klass._eval_eye(rows, cols)
+
+    @classmethod
+    def jordan_block(kls, size=None, eigenvalue=None, *, band='upper', **kwargs):
+        """Returns a Jordan block
+
+        Parameters
+        ==========
+
+        size : Integer, optional
+            Specifies the shape of the Jordan block matrix.
+
+        eigenvalue : Number or Symbol
+            Specifies the value for the main diagonal of the matrix.
+
+            .. note::
+                The keyword ``eigenval`` is also specified as an alias
+                of this keyword, but it is not recommended to use.
+
+                We may deprecate the alias in later release.
+
+        band : 'upper' or 'lower', optional
+            Specifies the position of the off-diagonal to put `1` s on.
+
+        cls : Matrix, optional
+            Specifies the matrix class of the output form.
+
+            If it is not specified, the class type where the method is
+            being executed on will be returned.
+
+        Returns
+        =======
+
+        Matrix
+            A Jordan block matrix.
+
+        Raises
+        ======
+
+        ValueError
+            If insufficient arguments are given for matrix size
+            specification, or no eigenvalue is given.
+
+        Examples
+        ========
+
+        Creating a default Jordan block:
+
+        >>> from sympy import Matrix
+        >>> from sympy.abc import x
+        >>> Matrix.jordan_block(4, x)
+        Matrix([
+        [x, 1, 0, 0],
+        [0, x, 1, 0],
+        [0, 0, x, 1],
+        [0, 0, 0, x]])
+
+        Creating an alternative Jordan block matrix where `1` is on
+        lower off-diagonal:
+
+        >>> Matrix.jordan_block(4, x, band='lower')
+        Matrix([
+        [x, 0, 0, 0],
+        [1, x, 0, 0],
+        [0, 1, x, 0],
+        [0, 0, 1, x]])
+
+        Creating a Jordan block with keyword arguments
+
+        >>> Matrix.jordan_block(size=4, eigenvalue=x)
+        Matrix([
+        [x, 1, 0, 0],
+        [0, x, 1, 0],
+        [0, 0, x, 1],
+        [0, 0, 0, x]])
+
+        References
+        ==========
+
+        .. [1] https://en.wikipedia.org/wiki/Jordan_matrix
+        """
+        klass = kwargs.pop('cls', kls)
+
+        eigenval = kwargs.get('eigenval', None)
+        if eigenvalue is None and eigenval is None:
+            raise ValueError("Must supply an eigenvalue")
+        elif eigenvalue != eigenval and None not in (eigenval, eigenvalue):
+            raise ValueError(
+                "Inconsistent values are given: 'eigenval'={}, "
+                "'eigenvalue'={}".format(eigenval, eigenvalue))
+        else:
+            if eigenval is not None:
+                eigenvalue = eigenval
+
+        if size is None:
+            raise ValueError("Must supply a matrix size")
+
+        size = as_int(size)
+        return klass._eval_jordan_block(size, eigenvalue, band)
+
+    @classmethod
+    def ones(kls, rows, cols=None, **kwargs):
+        """Returns a matrix of ones.
+
+        Parameters
+        ==========
+
+        rows : rows of the matrix
+        cols : cols of the matrix (if None, cols=rows)
+
+        kwargs
+        ======
+        cls : class of the returned matrix
+        """
+        if cols is None:
+            cols = rows
+        klass = kwargs.get('cls', kls)
+        rows, cols = as_int(rows), as_int(cols)
+
+        return klass._eval_ones(rows, cols)
+
+    @classmethod
+    def zeros(kls, rows, cols=None, **kwargs):
+        """Returns a matrix of zeros.
+
+        Parameters
+        ==========
+
+        rows : rows of the matrix
+        cols : cols of the matrix (if None, cols=rows)
+
+        kwargs
+        ======
+        cls : class of the returned matrix
+        """
+        if cols is None:
+            cols = rows
+        if rows < 0 or cols < 0:
+            raise ValueError("Cannot create a {} x {} matrix. "
+                             "Both dimensions must be positive".format(rows, cols))
+        klass = kwargs.get('cls', kls)
+        rows, cols = as_int(rows), as_int(cols)
+
+        return klass._eval_zeros(rows, cols)
+
+    @classmethod
+    def companion(kls, poly):
+        """Returns a companion matrix of a polynomial.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, Poly, Symbol, symbols
+        >>> x = Symbol('x')
+        >>> c0, c1, c2, c3, c4 = symbols('c0:5')
+        >>> p = Poly(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + x**5, x)
+        >>> Matrix.companion(p)
+        Matrix([
+        [0, 0, 0, 0, -c0],
+        [1, 0, 0, 0, -c1],
+        [0, 1, 0, 0, -c2],
+        [0, 0, 1, 0, -c3],
+        [0, 0, 0, 1, -c4]])
+        """
+        poly = kls._sympify(poly)
+        if not isinstance(poly, Poly):
+            raise ValueError("{} must be a Poly instance.".format(poly))
+        if not poly.is_monic:
+            raise ValueError("{} must be a monic polynomial.".format(poly))
+        if not poly.is_univariate:
+            raise ValueError(
+                "{} must be a univariate polynomial.".format(poly))
+
+        size = poly.degree()
+        if not size >= 1:
+            raise ValueError(
+                "{} must have degree not less than 1.".format(poly))
+
+        coeffs = poly.all_coeffs()
+        def entry(i, j):
+            if j == size - 1:
+                return -coeffs[-1 - i]
+            elif i == j + 1:
+                return kls.one
+            return kls.zero
+        return kls._new(size, size, entry)
+
+
+    @classmethod
+    def wilkinson(kls, n, **kwargs):
+        """Returns two square Wilkinson Matrix of size 2*n + 1
+        $W_{2n + 1}^-, W_{2n + 1}^+ =$ Wilkinson(n)
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> wminus, wplus = Matrix.wilkinson(3)
+        >>> wminus
+        Matrix([
+        [-3,  1,  0, 0, 0, 0, 0],
+        [ 1, -2,  1, 0, 0, 0, 0],
+        [ 0,  1, -1, 1, 0, 0, 0],
+        [ 0,  0,  1, 0, 1, 0, 0],
+        [ 0,  0,  0, 1, 1, 1, 0],
+        [ 0,  0,  0, 0, 1, 2, 1],
+        [ 0,  0,  0, 0, 0, 1, 3]])
+        >>> wplus
+        Matrix([
+        [3, 1, 0, 0, 0, 0, 0],
+        [1, 2, 1, 0, 0, 0, 0],
+        [0, 1, 1, 1, 0, 0, 0],
+        [0, 0, 1, 0, 1, 0, 0],
+        [0, 0, 0, 1, 1, 1, 0],
+        [0, 0, 0, 0, 1, 2, 1],
+        [0, 0, 0, 0, 0, 1, 3]])
+
+        References
+        ==========
+
+        .. [1] https://blogs.mathworks.com/cleve/2013/04/15/wilkinsons-matrices-2/
+        .. [2] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965, 662 pp.
+
+        """
+        klass = kwargs.get('cls', kls)
+        n = as_int(n)
+        return klass._eval_wilkinson(n)
+
+class MatrixProperties(MatrixRequired):
+    """Provides basic properties of a matrix."""
+
+    def _eval_atoms(self, *types):
+        result = set()
+        for i in self:
+            result.update(i.atoms(*types))
+        return result
+
+    def _eval_free_symbols(self):
+        return set().union(*(i.free_symbols for i in self if i))
+
+    def _eval_has(self, *patterns):
+        return any(a.has(*patterns) for a in self)
+
+    def _eval_is_anti_symmetric(self, simpfunc):
+        if not all(simpfunc(self[i, j] + self[j, i]).is_zero for i in range(self.rows) for j in range(self.cols)):
+            return False
+        return True
+
+    def _eval_is_diagonal(self):
+        for i in range(self.rows):
+            for j in range(self.cols):
+                if i != j and self[i, j]:
+                    return False
+        return True
+
+    # _eval_is_hermitian is called by some general SymPy
+    # routines and has a different *args signature.  Make
+    # sure the names don't clash by adding `_matrix_` in name.
+    def _eval_is_matrix_hermitian(self, simpfunc):
+        mat = self._new(self.rows, self.cols, lambda i, j: simpfunc(self[i, j] - self[j, i].conjugate()))
+        return mat.is_zero_matrix
+
+    def _eval_is_Identity(self) -> FuzzyBool:
+        def dirac(i, j):
+            if i == j:
+                return 1
+            return 0
+
+        return all(self[i, j] == dirac(i, j)
+                for i in range(self.rows)
+                for j in range(self.cols))
+
+    def _eval_is_lower_hessenberg(self):
+        return all(self[i, j].is_zero
+                   for i in range(self.rows)
+                   for j in range(i + 2, self.cols))
+
+    def _eval_is_lower(self):
+        return all(self[i, j].is_zero
+                   for i in range(self.rows)
+                   for j in range(i + 1, self.cols))
+
+    def _eval_is_symbolic(self):
+        return self.has(Symbol)
+
+    def _eval_is_symmetric(self, simpfunc):
+        mat = self._new(self.rows, self.cols, lambda i, j: simpfunc(self[i, j] - self[j, i]))
+        return mat.is_zero_matrix
+
+    def _eval_is_zero_matrix(self):
+        if any(i.is_zero == False for i in self):
+            return False
+        if any(i.is_zero is None for i in self):
+            return None
+        return True
+
+    def _eval_is_upper_hessenberg(self):
+        return all(self[i, j].is_zero
+                   for i in range(2, self.rows)
+                   for j in range(min(self.cols, (i - 1))))
+
+    def _eval_values(self):
+        return [i for i in self if not i.is_zero]
+
+    def _has_positive_diagonals(self):
+        diagonal_entries = (self[i, i] for i in range(self.rows))
+        return fuzzy_and(x.is_positive for x in diagonal_entries)
+
+    def _has_nonnegative_diagonals(self):
+        diagonal_entries = (self[i, i] for i in range(self.rows))
+        return fuzzy_and(x.is_nonnegative for x in diagonal_entries)
+
+    def atoms(self, *types):
+        """Returns the atoms that form the current object.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import Matrix
+        >>> Matrix([[x]])
+        Matrix([[x]])
+        >>> _.atoms()
+        {x}
+        >>> Matrix([[x, y], [y, x]])
+        Matrix([
+        [x, y],
+        [y, x]])
+        >>> _.atoms()
+        {x, y}
+        """
+
+        types = tuple(t if isinstance(t, type) else type(t) for t in types)
+        if not types:
+            types = (Atom,)
+        return self._eval_atoms(*types)
+
+    @property
+    def free_symbols(self):
+        """Returns the free symbols within the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import Matrix
+        >>> Matrix([[x], [1]]).free_symbols
+        {x}
+        """
+        return self._eval_free_symbols()
+
+    def has(self, *patterns):
+        """Test whether any subexpression matches any of the patterns.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, SparseMatrix, Float
+        >>> from sympy.abc import x, y
+        >>> A = Matrix(((1, x), (0.2, 3)))
+        >>> B = SparseMatrix(((1, x), (0.2, 3)))
+        >>> A.has(x)
+        True
+        >>> A.has(y)
+        False
+        >>> A.has(Float)
+        True
+        >>> B.has(x)
+        True
+        >>> B.has(y)
+        False
+        >>> B.has(Float)
+        True
+        """
+        return self._eval_has(*patterns)
+
+    def is_anti_symmetric(self, simplify=True):
+        """Check if matrix M is an antisymmetric matrix,
+        that is, M is a square matrix with all M[i, j] == -M[j, i].
+
+        When ``simplify=True`` (default), the sum M[i, j] + M[j, i] is
+        simplified before testing to see if it is zero. By default,
+        the SymPy simplify function is used. To use a custom function
+        set simplify to a function that accepts a single argument which
+        returns a simplified expression. To skip simplification, set
+        simplify to False but note that although this will be faster,
+        it may induce false negatives.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, symbols
+        >>> m = Matrix(2, 2, [0, 1, -1, 0])
+        >>> m
+        Matrix([
+        [ 0, 1],
+        [-1, 0]])
+        >>> m.is_anti_symmetric()
+        True
+        >>> x, y = symbols('x y')
+        >>> m = Matrix(2, 3, [0, 0, x, -y, 0, 0])
+        >>> m
+        Matrix([
+        [ 0, 0, x],
+        [-y, 0, 0]])
+        >>> m.is_anti_symmetric()
+        False
+
+        >>> from sympy.abc import x, y
+        >>> m = Matrix(3, 3, [0, x**2 + 2*x + 1, y,
+        ...                   -(x + 1)**2, 0, x*y,
+        ...                   -y, -x*y, 0])
+
+        Simplification of matrix elements is done by default so even
+        though two elements which should be equal and opposite would not
+        pass an equality test, the matrix is still reported as
+        anti-symmetric:
+
+        >>> m[0, 1] == -m[1, 0]
+        False
+        >>> m.is_anti_symmetric()
+        True
+
+        If ``simplify=False`` is used for the case when a Matrix is already
+        simplified, this will speed things up. Here, we see that without
+        simplification the matrix does not appear anti-symmetric:
+
+        >>> m.is_anti_symmetric(simplify=False)
+        False
+
+        But if the matrix were already expanded, then it would appear
+        anti-symmetric and simplification in the is_anti_symmetric routine
+        is not needed:
+
+        >>> m = m.expand()
+        >>> m.is_anti_symmetric(simplify=False)
+        True
+        """
+        # accept custom simplification
+        simpfunc = simplify
+        if not isfunction(simplify):
+            simpfunc = _simplify if simplify else lambda x: x
+
+        if not self.is_square:
+            return False
+        return self._eval_is_anti_symmetric(simpfunc)
+
+    def is_diagonal(self):
+        """Check if matrix is diagonal,
+        that is matrix in which the entries outside the main diagonal are all zero.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, diag
+        >>> m = Matrix(2, 2, [1, 0, 0, 2])
+        >>> m
+        Matrix([
+        [1, 0],
+        [0, 2]])
+        >>> m.is_diagonal()
+        True
+
+        >>> m = Matrix(2, 2, [1, 1, 0, 2])
+        >>> m
+        Matrix([
+        [1, 1],
+        [0, 2]])
+        >>> m.is_diagonal()
+        False
+
+        >>> m = diag(1, 2, 3)
+        >>> m
+        Matrix([
+        [1, 0, 0],
+        [0, 2, 0],
+        [0, 0, 3]])
+        >>> m.is_diagonal()
+        True
+
+        See Also
+        ========
+
+        is_lower
+        is_upper
+        sympy.matrices.matrices.MatrixEigen.is_diagonalizable
+        diagonalize
+        """
+        return self._eval_is_diagonal()
+
+    @property
+    def is_weakly_diagonally_dominant(self):
+        r"""Tests if the matrix is row weakly diagonally dominant.
+
+        Explanation
+        ===========
+
+        A $n, n$ matrix $A$ is row weakly diagonally dominant if
+
+        .. math::
+            \left|A_{i, i}\right| \ge \sum_{j = 0, j \neq i}^{n-1}
+            \left|A_{i, j}\right| \quad {\text{for all }}
+            i \in \{ 0, ..., n-1 \}
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix([[3, -2, 1], [1, -3, 2], [-1, 2, 4]])
+        >>> A.is_weakly_diagonally_dominant
+        True
+
+        >>> A = Matrix([[-2, 2, 1], [1, 3, 2], [1, -2, 0]])
+        >>> A.is_weakly_diagonally_dominant
+        False
+
+        >>> A = Matrix([[-4, 2, 1], [1, 6, 2], [1, -2, 5]])
+        >>> A.is_weakly_diagonally_dominant
+        True
+
+        Notes
+        =====
+
+        If you want to test whether a matrix is column diagonally
+        dominant, you can apply the test after transposing the matrix.
+        """
+        if not self.is_square:
+            return False
+
+        rows, cols = self.shape
+
+        def test_row(i):
+            summation = self.zero
+            for j in range(cols):
+                if i != j:
+                    summation += Abs(self[i, j])
+            return (Abs(self[i, i]) - summation).is_nonnegative
+
+        return fuzzy_and(test_row(i) for i in range(rows))
+
+    @property
+    def is_strongly_diagonally_dominant(self):
+        r"""Tests if the matrix is row strongly diagonally dominant.
+
+        Explanation
+        ===========
+
+        A $n, n$ matrix $A$ is row strongly diagonally dominant if
+
+        .. math::
+            \left|A_{i, i}\right| > \sum_{j = 0, j \neq i}^{n-1}
+            \left|A_{i, j}\right| \quad {\text{for all }}
+            i \in \{ 0, ..., n-1 \}
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix([[3, -2, 1], [1, -3, 2], [-1, 2, 4]])
+        >>> A.is_strongly_diagonally_dominant
+        False
+
+        >>> A = Matrix([[-2, 2, 1], [1, 3, 2], [1, -2, 0]])
+        >>> A.is_strongly_diagonally_dominant
+        False
+
+        >>> A = Matrix([[-4, 2, 1], [1, 6, 2], [1, -2, 5]])
+        >>> A.is_strongly_diagonally_dominant
+        True
+
+        Notes
+        =====
+
+        If you want to test whether a matrix is column diagonally
+        dominant, you can apply the test after transposing the matrix.
+        """
+        if not self.is_square:
+            return False
+
+        rows, cols = self.shape
+
+        def test_row(i):
+            summation = self.zero
+            for j in range(cols):
+                if i != j:
+                    summation += Abs(self[i, j])
+            return (Abs(self[i, i]) - summation).is_positive
+
+        return fuzzy_and(test_row(i) for i in range(rows))
+
+    @property
+    def is_hermitian(self):
+        """Checks if the matrix is Hermitian.
+
+        In a Hermitian matrix element i,j is the complex conjugate of
+        element j,i.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> from sympy import I
+        >>> from sympy.abc import x
+        >>> a = Matrix([[1, I], [-I, 1]])
+        >>> a
+        Matrix([
+        [ 1, I],
+        [-I, 1]])
+        >>> a.is_hermitian
+        True
+        >>> a[0, 0] = 2*I
+        >>> a.is_hermitian
+        False
+        >>> a[0, 0] = x
+        >>> a.is_hermitian
+        >>> a[0, 1] = a[1, 0]*I
+        >>> a.is_hermitian
+        False
+        """
+        if not self.is_square:
+            return False
+
+        return self._eval_is_matrix_hermitian(_simplify)
+
+    @property
+    def is_Identity(self) -> FuzzyBool:
+        if not self.is_square:
+            return False
+        return self._eval_is_Identity()
+
+    @property
+    def is_lower_hessenberg(self):
+        r"""Checks if the matrix is in the lower-Hessenberg form.
+
+        The lower hessenberg matrix has zero entries
+        above the first superdiagonal.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> a = Matrix([[1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]])
+        >>> a
+        Matrix([
+        [1, 2, 0, 0],
+        [5, 2, 3, 0],
+        [3, 4, 3, 7],
+        [5, 6, 1, 1]])
+        >>> a.is_lower_hessenberg
+        True
+
+        See Also
+        ========
+
+        is_upper_hessenberg
+        is_lower
+        """
+        return self._eval_is_lower_hessenberg()
+
+    @property
+    def is_lower(self):
+        """Check if matrix is a lower triangular matrix. True can be returned
+        even if the matrix is not square.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(2, 2, [1, 0, 0, 1])
+        >>> m
+        Matrix([
+        [1, 0],
+        [0, 1]])
+        >>> m.is_lower
+        True
+
+        >>> m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4, 0, 6, 6, 5])
+        >>> m
+        Matrix([
+        [0, 0, 0],
+        [2, 0, 0],
+        [1, 4, 0],
+        [6, 6, 5]])
+        >>> m.is_lower
+        True
+
+        >>> from sympy.abc import x, y
+        >>> m = Matrix(2, 2, [x**2 + y, y**2 + x, 0, x + y])
+        >>> m
+        Matrix([
+        [x**2 + y, x + y**2],
+        [       0,    x + y]])
+        >>> m.is_lower
+        False
+
+        See Also
+        ========
+
+        is_upper
+        is_diagonal
+        is_lower_hessenberg
+        """
+        return self._eval_is_lower()
+
+    @property
+    def is_square(self):
+        """Checks if a matrix is square.
+
+        A matrix is square if the number of rows equals the number of columns.
+        The empty matrix is square by definition, since the number of rows and
+        the number of columns are both zero.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> a = Matrix([[1, 2, 3], [4, 5, 6]])
+        >>> b = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+        >>> c = Matrix([])
+        >>> a.is_square
+        False
+        >>> b.is_square
+        True
+        >>> c.is_square
+        True
+        """
+        return self.rows == self.cols
+
+    def is_symbolic(self):
+        """Checks if any elements contain Symbols.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> from sympy.abc import x, y
+        >>> M = Matrix([[x, y], [1, 0]])
+        >>> M.is_symbolic()
+        True
+
+        """
+        return self._eval_is_symbolic()
+
+    def is_symmetric(self, simplify=True):
+        """Check if matrix is symmetric matrix,
+        that is square matrix and is equal to its transpose.
+
+        By default, simplifications occur before testing symmetry.
+        They can be skipped using 'simplify=False'; while speeding things a bit,
+        this may however induce false negatives.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(2, 2, [0, 1, 1, 2])
+        >>> m
+        Matrix([
+        [0, 1],
+        [1, 2]])
+        >>> m.is_symmetric()
+        True
+
+        >>> m = Matrix(2, 2, [0, 1, 2, 0])
+        >>> m
+        Matrix([
+        [0, 1],
+        [2, 0]])
+        >>> m.is_symmetric()
+        False
+
+        >>> m = Matrix(2, 3, [0, 0, 0, 0, 0, 0])
+        >>> m
+        Matrix([
+        [0, 0, 0],
+        [0, 0, 0]])
+        >>> m.is_symmetric()
+        False
+
+        >>> from sympy.abc import x, y
+        >>> m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2, 2, 0, y, 0, 3])
+        >>> m
+        Matrix([
+        [         1, x**2 + 2*x + 1, y],
+        [(x + 1)**2,              2, 0],
+        [         y,              0, 3]])
+        >>> m.is_symmetric()
+        True
+
+        If the matrix is already simplified, you may speed-up is_symmetric()
+        test by using 'simplify=False'.
+
+        >>> bool(m.is_symmetric(simplify=False))
+        False
+        >>> m1 = m.expand()
+        >>> m1.is_symmetric(simplify=False)
+        True
+        """
+        simpfunc = simplify
+        if not isfunction(simplify):
+            simpfunc = _simplify if simplify else lambda x: x
+
+        if not self.is_square:
+            return False
+
+        return self._eval_is_symmetric(simpfunc)
+
+    @property
+    def is_upper_hessenberg(self):
+        """Checks if the matrix is the upper-Hessenberg form.
+
+        The upper hessenberg matrix has zero entries
+        below the first subdiagonal.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> a = Matrix([[1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]])
+        >>> a
+        Matrix([
+        [1, 4, 2, 3],
+        [3, 4, 1, 7],
+        [0, 2, 3, 4],
+        [0, 0, 1, 3]])
+        >>> a.is_upper_hessenberg
+        True
+
+        See Also
+        ========
+
+        is_lower_hessenberg
+        is_upper
+        """
+        return self._eval_is_upper_hessenberg()
+
+    @property
+    def is_upper(self):
+        """Check if matrix is an upper triangular matrix. True can be returned
+        even if the matrix is not square.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(2, 2, [1, 0, 0, 1])
+        >>> m
+        Matrix([
+        [1, 0],
+        [0, 1]])
+        >>> m.is_upper
+        True
+
+        >>> m = Matrix(4, 3, [5, 1, 9, 0, 4, 6, 0, 0, 5, 0, 0, 0])
+        >>> m
+        Matrix([
+        [5, 1, 9],
+        [0, 4, 6],
+        [0, 0, 5],
+        [0, 0, 0]])
+        >>> m.is_upper
+        True
+
+        >>> m = Matrix(2, 3, [4, 2, 5, 6, 1, 1])
+        >>> m
+        Matrix([
+        [4, 2, 5],
+        [6, 1, 1]])
+        >>> m.is_upper
+        False
+
+        See Also
+        ========
+
+        is_lower
+        is_diagonal
+        is_upper_hessenberg
+        """
+        return all(self[i, j].is_zero
+                   for i in range(1, self.rows)
+                   for j in range(min(i, self.cols)))
+
+    @property
+    def is_zero_matrix(self):
+        """Checks if a matrix is a zero matrix.
+
+        A matrix is zero if every element is zero.  A matrix need not be square
+        to be considered zero.  The empty matrix is zero by the principle of
+        vacuous truth.  For a matrix that may or may not be zero (e.g.
+        contains a symbol), this will be None
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, zeros
+        >>> from sympy.abc import x
+        >>> a = Matrix([[0, 0], [0, 0]])
+        >>> b = zeros(3, 4)
+        >>> c = Matrix([[0, 1], [0, 0]])
+        >>> d = Matrix([])
+        >>> e = Matrix([[x, 0], [0, 0]])
+        >>> a.is_zero_matrix
+        True
+        >>> b.is_zero_matrix
+        True
+        >>> c.is_zero_matrix
+        False
+        >>> d.is_zero_matrix
+        True
+        >>> e.is_zero_matrix
+        """
+        return self._eval_is_zero_matrix()
+
+    def values(self):
+        """Return non-zero values of self."""
+        return self._eval_values()
+
+
+class MatrixOperations(MatrixRequired):
+    """Provides basic matrix shape and elementwise
+    operations.  Should not be instantiated directly."""
+
+    def _eval_adjoint(self):
+        return self.transpose().conjugate()
+
+    def _eval_applyfunc(self, f):
+        out = self._new(self.rows, self.cols, [f(x) for x in self])
+        return out
+
+    def _eval_as_real_imag(self):  # type: ignore
+        return (self.applyfunc(re), self.applyfunc(im))
+
+    def _eval_conjugate(self):
+        return self.applyfunc(lambda x: x.conjugate())
+
+    def _eval_permute_cols(self, perm):
+        # apply the permutation to a list
+        mapping = list(perm)
+
+        def entry(i, j):
+            return self[i, mapping[j]]
+
+        return self._new(self.rows, self.cols, entry)
+
+    def _eval_permute_rows(self, perm):
+        # apply the permutation to a list
+        mapping = list(perm)
+
+        def entry(i, j):
+            return self[mapping[i], j]
+
+        return self._new(self.rows, self.cols, entry)
+
+    def _eval_trace(self):
+        return sum(self[i, i] for i in range(self.rows))
+
+    def _eval_transpose(self):
+        return self._new(self.cols, self.rows, lambda i, j: self[j, i])
+
+    def adjoint(self):
+        """Conjugate transpose or Hermitian conjugation."""
+        return self._eval_adjoint()
+
+    def applyfunc(self, f):
+        """Apply a function to each element of the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> m = Matrix(2, 2, lambda i, j: i*2+j)
+        >>> m
+        Matrix([
+        [0, 1],
+        [2, 3]])
+        >>> m.applyfunc(lambda i: 2*i)
+        Matrix([
+        [0, 2],
+        [4, 6]])
+
+        """
+        if not callable(f):
+            raise TypeError("`f` must be callable.")
+
+        return self._eval_applyfunc(f)
+
+    def as_real_imag(self, deep=True, **hints):
+        """Returns a tuple containing the (real, imaginary) part of matrix."""
+        # XXX: Ignoring deep and hints...
+        return self._eval_as_real_imag()
+
+    def conjugate(self):
+        """Return the by-element conjugation.
+
+        Examples
+        ========
+
+        >>> from sympy import SparseMatrix, I
+        >>> a = SparseMatrix(((1, 2 + I), (3, 4), (I, -I)))
+        >>> a
+        Matrix([
+        [1, 2 + I],
+        [3,     4],
+        [I,    -I]])
+        >>> a.C
+        Matrix([
+        [ 1, 2 - I],
+        [ 3,     4],
+        [-I,     I]])
+
+        See Also
+        ========
+
+        transpose: Matrix transposition
+        H: Hermite conjugation
+        sympy.matrices.matrices.MatrixBase.D: Dirac conjugation
+        """
+        return self._eval_conjugate()
+
+    def doit(self, **hints):
+        return self.applyfunc(lambda x: x.doit(**hints))
+
+    def evalf(self, n=15, subs=None, maxn=100, chop=False, strict=False, quad=None, verbose=False):
+        """Apply evalf() to each element of self."""
+        options = {'subs':subs, 'maxn':maxn, 'chop':chop, 'strict':strict,
+                'quad':quad, 'verbose':verbose}
+        return self.applyfunc(lambda i: i.evalf(n, **options))
+
+    def expand(self, deep=True, modulus=None, power_base=True, power_exp=True,
+               mul=True, log=True, multinomial=True, basic=True, **hints):
+        """Apply core.function.expand to each entry of the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import Matrix
+        >>> Matrix(1, 1, [x*(x+1)])
+        Matrix([[x*(x + 1)]])
+        >>> _.expand()
+        Matrix([[x**2 + x]])
+
+        """
+        return self.applyfunc(lambda x: x.expand(
+            deep, modulus, power_base, power_exp, mul, log, multinomial, basic,
+            **hints))
+
+    @property
+    def H(self):
+        """Return Hermite conjugate.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, I
+        >>> m = Matrix((0, 1 + I, 2, 3))
+        >>> m
+        Matrix([
+        [    0],
+        [1 + I],
+        [    2],
+        [    3]])
+        >>> m.H
+        Matrix([[0, 1 - I, 2, 3]])
+
+        See Also
+        ========
+
+        conjugate: By-element conjugation
+        sympy.matrices.matrices.MatrixBase.D: Dirac conjugation
+        """
+        return self.T.C
+
+    def permute(self, perm, orientation='rows', direction='forward'):
+        r"""Permute the rows or columns of a matrix by the given list of
+        swaps.
+
+        Parameters
+        ==========
+
+        perm : Permutation, list, or list of lists
+            A representation for the permutation.
+
+            If it is ``Permutation``, it is used directly with some
+            resizing with respect to the matrix size.
+
+            If it is specified as list of lists,
+            (e.g., ``[[0, 1], [0, 2]]``), then the permutation is formed
+            from applying the product of cycles. The direction how the
+            cyclic product is applied is described in below.
+
+            If it is specified as a list, the list should represent
+            an array form of a permutation. (e.g., ``[1, 2, 0]``) which
+            would would form the swapping function
+            `0 \mapsto 1, 1 \mapsto 2, 2\mapsto 0`.
+
+        orientation : 'rows', 'cols'
+            A flag to control whether to permute the rows or the columns
+
+        direction : 'forward', 'backward'
+            A flag to control whether to apply the permutations from
+            the start of the list first, or from the back of the list
+            first.
+
+            For example, if the permutation specification is
+            ``[[0, 1], [0, 2]]``,
+
+            If the flag is set to ``'forward'``, the cycle would be
+            formed as `0 \mapsto 2, 2 \mapsto 1, 1 \mapsto 0`.
+
+            If the flag is set to ``'backward'``, the cycle would be
+            formed as `0 \mapsto 1, 1 \mapsto 2, 2 \mapsto 0`.
+
+            If the argument ``perm`` is not in a form of list of lists,
+            this flag takes no effect.
+
+        Examples
+        ========
+
+        >>> from sympy import eye
+        >>> M = eye(3)
+        >>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='forward')
+        Matrix([
+        [0, 0, 1],
+        [1, 0, 0],
+        [0, 1, 0]])
+
+        >>> from sympy import eye
+        >>> M = eye(3)
+        >>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='backward')
+        Matrix([
+        [0, 1, 0],
+        [0, 0, 1],
+        [1, 0, 0]])
+
+        Notes
+        =====
+
+        If a bijective function
+        `\sigma : \mathbb{N}_0 \rightarrow \mathbb{N}_0` denotes the
+        permutation.
+
+        If the matrix `A` is the matrix to permute, represented as
+        a horizontal or a vertical stack of vectors:
+
+        .. math::
+            A =
+            \begin{bmatrix}
+            a_0 \\ a_1 \\ \vdots \\ a_{n-1}
+            \end{bmatrix} =
+            \begin{bmatrix}
+            \alpha_0 & \alpha_1 & \cdots & \alpha_{n-1}
+            \end{bmatrix}
+
+        If the matrix `B` is the result, the permutation of matrix rows
+        is defined as:
+
+        .. math::
+            B := \begin{bmatrix}
+            a_{\sigma(0)} \\ a_{\sigma(1)} \\ \vdots \\ a_{\sigma(n-1)}
+            \end{bmatrix}
+
+        And the permutation of matrix columns is defined as:
+
+        .. math::
+            B := \begin{bmatrix}
+            \alpha_{\sigma(0)} & \alpha_{\sigma(1)} &
+            \cdots & \alpha_{\sigma(n-1)}
+            \end{bmatrix}
+        """
+        from sympy.combinatorics import Permutation
+
+        # allow british variants and `columns`
+        if direction == 'forwards':
+            direction = 'forward'
+        if direction == 'backwards':
+            direction = 'backward'
+        if orientation == 'columns':
+            orientation = 'cols'
+
+        if direction not in ('forward', 'backward'):
+            raise TypeError("direction='{}' is an invalid kwarg. "
+                            "Try 'forward' or 'backward'".format(direction))
+        if orientation not in ('rows', 'cols'):
+            raise TypeError("orientation='{}' is an invalid kwarg. "
+                            "Try 'rows' or 'cols'".format(orientation))
+
+        if not isinstance(perm, (Permutation, Iterable)):
+            raise ValueError(
+                "{} must be a list, a list of lists, "
+                "or a SymPy permutation object.".format(perm))
+
+        # ensure all swaps are in range
+        max_index = self.rows if orientation == 'rows' else self.cols
+        if not all(0 <= t <= max_index for t in flatten(list(perm))):
+            raise IndexError("`swap` indices out of range.")
+
+        if perm and not isinstance(perm, Permutation) and \
+            isinstance(perm[0], Iterable):
+            if direction == 'forward':
+                perm = list(reversed(perm))
+            perm = Permutation(perm, size=max_index+1)
+        else:
+            perm = Permutation(perm, size=max_index+1)
+
+        if orientation == 'rows':
+            return self._eval_permute_rows(perm)
+        if orientation == 'cols':
+            return self._eval_permute_cols(perm)
+
+    def permute_cols(self, swaps, direction='forward'):
+        """Alias for
+        ``self.permute(swaps, orientation='cols', direction=direction)``
+
+        See Also
+        ========
+
+        permute
+        """
+        return self.permute(swaps, orientation='cols', direction=direction)
+
+    def permute_rows(self, swaps, direction='forward'):
+        """Alias for
+        ``self.permute(swaps, orientation='rows', direction=direction)``
+
+        See Also
+        ========
+
+        permute
+        """
+        return self.permute(swaps, orientation='rows', direction=direction)
+
+    def refine(self, assumptions=True):
+        """Apply refine to each element of the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy import Symbol, Matrix, Abs, sqrt, Q
+        >>> x = Symbol('x')
+        >>> Matrix([[Abs(x)**2, sqrt(x**2)],[sqrt(x**2), Abs(x)**2]])
+        Matrix([
+        [ Abs(x)**2, sqrt(x**2)],
+        [sqrt(x**2),  Abs(x)**2]])
+        >>> _.refine(Q.real(x))
+        Matrix([
+        [  x**2, Abs(x)],
+        [Abs(x),   x**2]])
+
+        """
+        return self.applyfunc(lambda x: refine(x, assumptions))
+
+    def replace(self, F, G, map=False, simultaneous=True, exact=None):
+        """Replaces Function F in Matrix entries with Function G.
+
+        Examples
+        ========
+
+        >>> from sympy import symbols, Function, Matrix
+        >>> F, G = symbols('F, G', cls=Function)
+        >>> M = Matrix(2, 2, lambda i, j: F(i+j)) ; M
+        Matrix([
+        [F(0), F(1)],
+        [F(1), F(2)]])
+        >>> N = M.replace(F,G)
+        >>> N
+        Matrix([
+        [G(0), G(1)],
+        [G(1), G(2)]])
+        """
+        return self.applyfunc(
+            lambda x: x.replace(F, G, map=map, simultaneous=simultaneous, exact=exact))
+
+    def rot90(self, k=1):
+        """Rotates Matrix by 90 degrees
+
+        Parameters
+        ==========
+
+        k : int
+            Specifies how many times the matrix is rotated by 90 degrees
+            (clockwise when positive, counter-clockwise when negative).
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix, symbols
+        >>> A = Matrix(2, 2, symbols('a:d'))
+        >>> A
+        Matrix([
+        [a, b],
+        [c, d]])
+
+        Rotating the matrix clockwise one time:
+
+        >>> A.rot90(1)
+        Matrix([
+        [c, a],
+        [d, b]])
+
+        Rotating the matrix anticlockwise two times:
+
+        >>> A.rot90(-2)
+        Matrix([
+        [d, c],
+        [b, a]])
+        """
+
+        mod = k%4
+        if mod == 0:
+            return self
+        if mod == 1:
+            return self[::-1, ::].T
+        if mod == 2:
+            return self[::-1, ::-1]
+        if mod == 3:
+            return self[::, ::-1].T
+
+    def simplify(self, **kwargs):
+        """Apply simplify to each element of the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import SparseMatrix, sin, cos
+        >>> SparseMatrix(1, 1, [x*sin(y)**2 + x*cos(y)**2])
+        Matrix([[x*sin(y)**2 + x*cos(y)**2]])
+        >>> _.simplify()
+        Matrix([[x]])
+        """
+        return self.applyfunc(lambda x: x.simplify(**kwargs))
+
+    def subs(self, *args, **kwargs):  # should mirror core.basic.subs
+        """Return a new matrix with subs applied to each entry.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import SparseMatrix, Matrix
+        >>> SparseMatrix(1, 1, [x])
+        Matrix([[x]])
+        >>> _.subs(x, y)
+        Matrix([[y]])
+        >>> Matrix(_).subs(y, x)
+        Matrix([[x]])
+        """
+
+        if len(args) == 1 and  not isinstance(args[0], (dict, set)) and iter(args[0]) and not is_sequence(args[0]):
+            args = (list(args[0]),)
+
+        return self.applyfunc(lambda x: x.subs(*args, **kwargs))
+
+    def trace(self):
+        """
+        Returns the trace of a square matrix i.e. the sum of the
+        diagonal elements.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix(2, 2, [1, 2, 3, 4])
+        >>> A.trace()
+        5
+
+        """
+        if self.rows != self.cols:
+            raise NonSquareMatrixError()
+        return self._eval_trace()
+
+    def transpose(self):
+        """
+        Returns the transpose of the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix(2, 2, [1, 2, 3, 4])
+        >>> A.transpose()
+        Matrix([
+        [1, 3],
+        [2, 4]])
+
+        >>> from sympy import Matrix, I
+        >>> m=Matrix(((1, 2+I), (3, 4)))
+        >>> m
+        Matrix([
+        [1, 2 + I],
+        [3,     4]])
+        >>> m.transpose()
+        Matrix([
+        [    1, 3],
+        [2 + I, 4]])
+        >>> m.T == m.transpose()
+        True
+
+        See Also
+        ========
+
+        conjugate: By-element conjugation
+
+        """
+        return self._eval_transpose()
+
+    @property
+    def T(self):
+        '''Matrix transposition'''
+        return self.transpose()
+
+    @property
+    def C(self):
+        '''By-element conjugation'''
+        return self.conjugate()
+
+    def n(self, *args, **kwargs):
+        """Apply evalf() to each element of self."""
+        return self.evalf(*args, **kwargs)
+
+    def xreplace(self, rule):  # should mirror core.basic.xreplace
+        """Return a new matrix with xreplace applied to each entry.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import SparseMatrix, Matrix
+        >>> SparseMatrix(1, 1, [x])
+        Matrix([[x]])
+        >>> _.xreplace({x: y})
+        Matrix([[y]])
+        >>> Matrix(_).xreplace({y: x})
+        Matrix([[x]])
+        """
+        return self.applyfunc(lambda x: x.xreplace(rule))
+
+    def _eval_simplify(self, **kwargs):
+        # XXX: We can't use self.simplify here as mutable subclasses will
+        # override simplify and have it return None
+        return MatrixOperations.simplify(self, **kwargs)
+
+    def _eval_trigsimp(self, **opts):
+        from sympy.simplify.trigsimp import trigsimp
+        return self.applyfunc(lambda x: trigsimp(x, **opts))
+
+    def upper_triangular(self, k=0):
+        """Return the elements on and above the kth diagonal of a matrix.
+        If k is not specified then simply returns upper-triangular portion
+        of a matrix
+
+        Examples
+        ========
+
+        >>> from sympy import ones
+        >>> A = ones(4)
+        >>> A.upper_triangular()
+        Matrix([
+        [1, 1, 1, 1],
+        [0, 1, 1, 1],
+        [0, 0, 1, 1],
+        [0, 0, 0, 1]])
+
+        >>> A.upper_triangular(2)
+        Matrix([
+        [0, 0, 1, 1],
+        [0, 0, 0, 1],
+        [0, 0, 0, 0],
+        [0, 0, 0, 0]])
+
+        >>> A.upper_triangular(-1)
+        Matrix([
+        [1, 1, 1, 1],
+        [1, 1, 1, 1],
+        [0, 1, 1, 1],
+        [0, 0, 1, 1]])
+
+        """
+
+        def entry(i, j):
+            return self[i, j] if i + k <= j else self.zero
+
+        return self._new(self.rows, self.cols, entry)
+
+
+    def lower_triangular(self, k=0):
+        """Return the elements on and below the kth diagonal of a matrix.
+        If k is not specified then simply returns lower-triangular portion
+        of a matrix
+
+        Examples
+        ========
+
+        >>> from sympy import ones
+        >>> A = ones(4)
+        >>> A.lower_triangular()
+        Matrix([
+        [1, 0, 0, 0],
+        [1, 1, 0, 0],
+        [1, 1, 1, 0],
+        [1, 1, 1, 1]])
+
+        >>> A.lower_triangular(-2)
+        Matrix([
+        [0, 0, 0, 0],
+        [0, 0, 0, 0],
+        [1, 0, 0, 0],
+        [1, 1, 0, 0]])
+
+        >>> A.lower_triangular(1)
+        Matrix([
+        [1, 1, 0, 0],
+        [1, 1, 1, 0],
+        [1, 1, 1, 1],
+        [1, 1, 1, 1]])
+
+        """
+
+        def entry(i, j):
+            return self[i, j] if i + k >= j else self.zero
+
+        return self._new(self.rows, self.cols, entry)
+
+
+
+class MatrixArithmetic(MatrixRequired):
+    """Provides basic matrix arithmetic operations.
+    Should not be instantiated directly."""
+
+    _op_priority = 10.01
+
+    def _eval_Abs(self):
+        return self._new(self.rows, self.cols, lambda i, j: Abs(self[i, j]))
+
+    def _eval_add(self, other):
+        return self._new(self.rows, self.cols,
+                         lambda i, j: self[i, j] + other[i, j])
+
+    def _eval_matrix_mul(self, other):
+        def entry(i, j):
+            vec = [self[i,k]*other[k,j] for k in range(self.cols)]
+            try:
+                return Add(*vec)
+            except (TypeError, SympifyError):
+                # Some matrices don't work with `sum` or `Add`
+                # They don't work with `sum` because `sum` tries to add `0`
+                # Fall back to a safe way to multiply if the `Add` fails.
+                return reduce(lambda a, b: a + b, vec)
+
+        return self._new(self.rows, other.cols, entry)
+
+    def _eval_matrix_mul_elementwise(self, other):
+        return self._new(self.rows, self.cols, lambda i, j: self[i,j]*other[i,j])
+
+    def _eval_matrix_rmul(self, other):
+        def entry(i, j):
+            return sum(other[i,k]*self[k,j] for k in range(other.cols))
+        return self._new(other.rows, self.cols, entry)
+
+    def _eval_pow_by_recursion(self, num):
+        if num == 1:
+            return self
+
+        if num % 2 == 1:
+            a, b = self, self._eval_pow_by_recursion(num - 1)
+        else:
+            a = b = self._eval_pow_by_recursion(num // 2)
+
+        return a.multiply(b)
+
+    def _eval_pow_by_cayley(self, exp):
+        from sympy.discrete.recurrences import linrec_coeffs
+        row = self.shape[0]
+        p = self.charpoly()
+
+        coeffs = (-p).all_coeffs()[1:]
+        coeffs = linrec_coeffs(coeffs, exp)
+        new_mat = self.eye(row)
+        ans = self.zeros(row)
+
+        for i in range(row):
+            ans += coeffs[i]*new_mat
+            new_mat *= self
+
+        return ans
+
+    def _eval_pow_by_recursion_dotprodsimp(self, num, prevsimp=None):
+        if prevsimp is None:
+            prevsimp = [True]*len(self)
+
+        if num == 1:
+            return self
+
+        if num % 2 == 1:
+            a, b = self, self._eval_pow_by_recursion_dotprodsimp(num - 1,
+                    prevsimp=prevsimp)
+        else:
+            a = b = self._eval_pow_by_recursion_dotprodsimp(num // 2,
+                    prevsimp=prevsimp)
+
+        m     = a.multiply(b, dotprodsimp=False)
+        lenm  = len(m)
+        elems = [None]*lenm
+
+        for i in range(lenm):
+            if prevsimp[i]:
+                elems[i], prevsimp[i] = _dotprodsimp(m[i], withsimp=True)
+            else:
+                elems[i] = m[i]
+
+        return m._new(m.rows, m.cols, elems)
+
+    def _eval_scalar_mul(self, other):
+        return self._new(self.rows, self.cols, lambda i, j: self[i,j]*other)
+
+    def _eval_scalar_rmul(self, other):
+        return self._new(self.rows, self.cols, lambda i, j: other*self[i,j])
+
+    def _eval_Mod(self, other):
+        return self._new(self.rows, self.cols, lambda i, j: Mod(self[i, j], other))
+
+    # Python arithmetic functions
+    def __abs__(self):
+        """Returns a new matrix with entry-wise absolute values."""
+        return self._eval_Abs()
+
+    @call_highest_priority('__radd__')
+    def __add__(self, other):
+        """Return self + other, raising ShapeError if shapes do not match."""
+        if isinstance(other, NDimArray): # Matrix and array addition is currently not implemented
+            return NotImplemented
+        other = _matrixify(other)
+        # matrix-like objects can have shapes.  This is
+        # our first sanity check.
+        if hasattr(other, 'shape'):
+            if self.shape != other.shape:
+                raise ShapeError("Matrix size mismatch: %s + %s" % (
+                    self.shape, other.shape))
+
+        # honest SymPy matrices defer to their class's routine
+        if getattr(other, 'is_Matrix', False):
+            # call the highest-priority class's _eval_add
+            a, b = self, other
+            if a.__class__ != classof(a, b):
+                b, a = a, b
+            return a._eval_add(b)
+        # Matrix-like objects can be passed to CommonMatrix routines directly.
+        if getattr(other, 'is_MatrixLike', False):
+            return MatrixArithmetic._eval_add(self, other)
+
+        raise TypeError('cannot add %s and %s' % (type(self), type(other)))
+
+    @call_highest_priority('__rtruediv__')
+    def __truediv__(self, other):
+        return self * (self.one / other)
+
+    @call_highest_priority('__rmatmul__')
+    def __matmul__(self, other):
+        other = _matrixify(other)
+        if not getattr(other, 'is_Matrix', False) and not getattr(other, 'is_MatrixLike', False):
+            return NotImplemented
+
+        return self.__mul__(other)
+
+    def __mod__(self, other):
+        return self.applyfunc(lambda x: x % other)
+
+    @call_highest_priority('__rmul__')
+    def __mul__(self, other):
+        """Return self*other where other is either a scalar or a matrix
+        of compatible dimensions.
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix([[1, 2, 3], [4, 5, 6]])
+        >>> 2*A == A*2 == Matrix([[2, 4, 6], [8, 10, 12]])
+        True
+        >>> B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+        >>> A*B
+        Matrix([
+        [30, 36, 42],
+        [66, 81, 96]])
+        >>> B*A
+        Traceback (most recent call last):
+        ...
+        ShapeError: Matrices size mismatch.
+        >>>
+
+        See Also
+        ========
+
+        matrix_multiply_elementwise
+        """
+
+        return self.multiply(other)
+
+    def multiply(self, other, dotprodsimp=None):
+        """Same as __mul__() but with optional simplification.
+
+        Parameters
+        ==========
+
+        dotprodsimp : bool, optional
+            Specifies whether intermediate term algebraic simplification is used
+            during matrix multiplications to control expression blowup and thus
+            speed up calculation. Default is off.
+        """
+
+        isimpbool = _get_intermediate_simp_bool(False, dotprodsimp)
+        other = _matrixify(other)
+        # matrix-like objects can have shapes.  This is
+        # our first sanity check. Double check other is not explicitly not a Matrix.
+        if (hasattr(other, 'shape') and len(other.shape) == 2 and
+            (getattr(other, 'is_Matrix', True) or
+             getattr(other, 'is_MatrixLike', True))):
+            if self.shape[1] != other.shape[0]:
+                raise ShapeError("Matrix size mismatch: %s * %s." % (
+                    self.shape, other.shape))
+
+        # honest SymPy matrices defer to their class's routine
+        if getattr(other, 'is_Matrix', False):
+            m = self._eval_matrix_mul(other)
+            if isimpbool:
+                return m._new(m.rows, m.cols, [_dotprodsimp(e) for e in m])
+            return m
+
+        # Matrix-like objects can be passed to CommonMatrix routines directly.
+        if getattr(other, 'is_MatrixLike', False):
+            return MatrixArithmetic._eval_matrix_mul(self, other)
+
+        # if 'other' is not iterable then scalar multiplication.
+        if not isinstance(other, Iterable):
+            try:
+                return self._eval_scalar_mul(other)
+            except TypeError:
+                pass
+
+        return NotImplemented
+
+    def multiply_elementwise(self, other):
+        """Return the Hadamard product (elementwise product) of A and B
+
+        Examples
+        ========
+
+        >>> from sympy import Matrix
+        >>> A = Matrix([[0, 1, 2], [3, 4, 5]])
+        >>> B = Matrix([[1, 10, 100], [100, 10, 1]])
+        >>> A.multiply_elementwise(B)
+        Matrix([
+        [  0, 10, 200],
+        [300, 40,   5]])
+
+        See Also
+        ========
+
+        sympy.matrices.matrices.MatrixBase.cross
+        sympy.matrices.matrices.MatrixBase.dot
+        multiply
+        """
+        if self.shape != other.shape:
+            raise ShapeError("Matrix shapes must agree {} != {}".format(self.shape, other.shape))
+
+        return self._eval_matrix_mul_elementwise(other)
+
+    def __neg__(self):
+        return self._eval_scalar_mul(-1)
+
+    @call_highest_priority('__rpow__')
+    def __pow__(self, exp):
+        """Return self**exp a scalar or symbol."""
+
+        return self.pow(exp)
+
+
+    def pow(self, exp, method=None):
+        r"""Return self**exp a scalar or symbol.
+
+        Parameters
+        ==========
+
+        method : multiply, mulsimp, jordan, cayley
+            If multiply then it returns exponentiation using recursion.
+            If jordan then Jordan form exponentiation will be used.
+            If cayley then the exponentiation is done using Cayley-Hamilton
+            theorem.
+            If mulsimp then the exponentiation is done using recursion
+            with dotprodsimp. This specifies whether intermediate term
+            algebraic simplification is used during naive matrix power to
+            control expression blowup and thus speed up calculation.
+            If None, then it heuristically decides which method to use.
+
+        """
+
+        if method is not None and method not in ['multiply', 'mulsimp', 'jordan', 'cayley']:
+            raise TypeError('No such method')
+        if self.rows != self.cols:
+            raise NonSquareMatrixError()
+        a = self
+        jordan_pow = getattr(a, '_matrix_pow_by_jordan_blocks', None)
+        exp = sympify(exp)
+
+        if exp.is_zero:
+            return a._new(a.rows, a.cols, lambda i, j: int(i == j))
+        if exp == 1:
+            return a
+
+        diagonal = getattr(a, 'is_diagonal', None)
+        if diagonal is not None and diagonal():
+            return a._new(a.rows, a.cols, lambda i, j: a[i,j]**exp if i == j else 0)
+
+        if exp.is_Number and exp % 1 == 0:
+            if a.rows == 1:
+                return a._new([[a[0]**exp]])
+            if exp < 0:
+                exp = -exp
+                a = a.inv()
+        # When certain conditions are met,
+        # Jordan block algorithm is faster than
+        # computation by recursion.
+        if method == 'jordan':
+            try:
+                return jordan_pow(exp)
+            except MatrixError:
+                if method == 'jordan':
+                    raise
+
+        elif method == 'cayley':
+            if not exp.is_Number or exp % 1 != 0:
+                raise ValueError("cayley method is only valid for integer powers")
+            return a._eval_pow_by_cayley(exp)
+
+        elif method == "mulsimp":
+            if not exp.is_Number or exp % 1 != 0:
+                raise ValueError("mulsimp method is only valid for integer powers")
+            return a._eval_pow_by_recursion_dotprodsimp(exp)
+
+        elif method == "multiply":
+            if not exp.is_Number or exp % 1 != 0:
+                raise ValueError("multiply method is only valid for integer powers")
+            return a._eval_pow_by_recursion(exp)
+
+        elif method is None and exp.is_Number and exp % 1 == 0:
+            # Decide heuristically which method to apply
+            if a.rows == 2 and exp > 100000:
+                return jordan_pow(exp)
+            elif _get_intermediate_simp_bool(True, None):
+                return a._eval_pow_by_recursion_dotprodsimp(exp)
+            elif exp > 10000:
+                return a._eval_pow_by_cayley(exp)
+            else:
+                return a._eval_pow_by_recursion(exp)
+
+        if jordan_pow:
+            try:
+                return jordan_pow(exp)
+            except NonInvertibleMatrixError:
+                # Raised by jordan_pow on zero determinant matrix unless exp is
+                # definitely known to be a non-negative integer.
+                # Here we raise if n is definitely not a non-negative integer
+                # but otherwise we can leave this as an unevaluated MatPow.
+                if exp.is_integer is False or exp.is_nonnegative is False:
+                    raise
+
+        from sympy.matrices.expressions import MatPow
+        return MatPow(a, exp)
+
+    @call_highest_priority('__add__')
+    def __radd__(self, other):
+        return self + other
+
+    @call_highest_priority('__matmul__')
+    def __rmatmul__(self, other):
+        other = _matrixify(other)
+        if not getattr(other, 'is_Matrix', False) and not getattr(other, 'is_MatrixLike', False):
+            return NotImplemented
+
+        return self.__rmul__(other)
+
+    @call_highest_priority('__mul__')
+    def __rmul__(self, other):
+        return self.rmultiply(other)
+
+    def rmultiply(self, other, dotprodsimp=None):
+        """Same as __rmul__() but with optional simplification.
+
+        Parameters
+        ==========
+
+        dotprodsimp : bool, optional
+            Specifies whether intermediate term algebraic simplification is used
+            during matrix multiplications to control expression blowup and thus
+            speed up calculation. Default is off.
+        """
+        isimpbool = _get_intermediate_simp_bool(False, dotprodsimp)
+        other = _matrixify(other)
+        # matrix-like objects can have shapes.  This is
+        # our first sanity check. Double check other is not explicitly not a Matrix.
+        if (hasattr(other, 'shape') and len(other.shape) == 2 and
+            (getattr(other, 'is_Matrix', True) or
+             getattr(other, 'is_MatrixLike', True))):
+            if self.shape[0] != other.shape[1]:
+                raise ShapeError("Matrix size mismatch.")
+
+        # honest SymPy matrices defer to their class's routine
+        if getattr(other, 'is_Matrix', False):
+            m = self._eval_matrix_rmul(other)
+            if isimpbool:
+                return m._new(m.rows, m.cols, [_dotprodsimp(e) for e in m])
+            return m
+        # Matrix-like objects can be passed to CommonMatrix routines directly.
+        if getattr(other, 'is_MatrixLike', False):
+            return MatrixArithmetic._eval_matrix_rmul(self, other)
+
+        # if 'other' is not iterable then scalar multiplication.
+        if not isinstance(other, Iterable):
+            try:
+                return self._eval_scalar_rmul(other)
+            except TypeError:
+                pass
+
+        return NotImplemented
+
+    @call_highest_priority('__sub__')
+    def __rsub__(self, a):
+        return (-self) + a
+
+    @call_highest_priority('__rsub__')
+    def __sub__(self, a):
+        return self + (-a)
+
+class MatrixCommon(MatrixArithmetic, MatrixOperations, MatrixProperties,
+                  MatrixSpecial, MatrixShaping):
+    """All common matrix operations including basic arithmetic, shaping,
+    and special matrices like `zeros`, and `eye`."""
+    _diff_wrt = True  # type: bool
+
+
+class _MinimalMatrix:
+    """Class providing the minimum functionality
+    for a matrix-like object and implementing every method
+    required for a `MatrixRequired`.  This class does not have everything
+    needed to become a full-fledged SymPy object, but it will satisfy the
+    requirements of anything inheriting from `MatrixRequired`.  If you wish
+    to make a specialized matrix type, make sure to implement these
+    methods and properties with the exception of `__init__` and `__repr__`
+    which are included for convenience."""
+
+    is_MatrixLike = True
+    _sympify = staticmethod(sympify)
+    _class_priority = 3
+    zero = S.Zero
+    one = S.One
+
+    is_Matrix = True
+    is_MatrixExpr = False
+
+    @classmethod
+    def _new(cls, *args, **kwargs):
+        return cls(*args, **kwargs)
+
+    def __init__(self, rows, cols=None, mat=None, copy=False):
+        if isfunction(mat):
+            # if we passed in a function, use that to populate the indices
+            mat = [mat(i, j) for i in range(rows) for j in range(cols)]
+        if cols is None and mat is None:
+            mat = rows
+        rows, cols = getattr(mat, 'shape', (rows, cols))
+        try:
+            # if we passed in a list of lists, flatten it and set the size
+            if cols is None and mat is None:
+                mat = rows
+            cols = len(mat[0])
+            rows = len(mat)
+            mat = [x for l in mat for x in l]
+        except (IndexError, TypeError):
+            pass
+        self.mat = tuple(self._sympify(x) for x in mat)
+        self.rows, self.cols = rows, cols
+        if self.rows is None or self.cols is None:
+            raise NotImplementedError("Cannot initialize matrix with given parameters")
+
+    def __getitem__(self, key):
+        def _normalize_slices(row_slice, col_slice):
+            """Ensure that row_slice and col_slice do not have
+            `None` in their arguments.  Any integers are converted
+            to slices of length 1"""
+            if not isinstance(row_slice, slice):
+                row_slice = slice(row_slice, row_slice + 1, None)
+            row_slice = slice(*row_slice.indices(self.rows))
+
+            if not isinstance(col_slice, slice):
+                col_slice = slice(col_slice, col_slice + 1, None)
+            col_slice = slice(*col_slice.indices(self.cols))
+
+            return (row_slice, col_slice)
+
+        def _coord_to_index(i, j):
+            """Return the index in _mat corresponding
+            to the (i,j) position in the matrix. """
+            return i * self.cols + j
+
+        if isinstance(key, tuple):
+            i, j = key
+            if isinstance(i, slice) or isinstance(j, slice):
+                # if the coordinates are not slices, make them so
+                # and expand the slices so they don't contain `None`
+                i, j = _normalize_slices(i, j)
+
+                rowsList, colsList = list(range(self.rows))[i], \
+                                     list(range(self.cols))[j]
+                indices = (i * self.cols + j for i in rowsList for j in
+                           colsList)
+                return self._new(len(rowsList), len(colsList),
+                                 [self.mat[i] for i in indices])
+
+            # if the key is a tuple of ints, change
+            # it to an array index
+            key = _coord_to_index(i, j)
+        return self.mat[key]
+
+    def __eq__(self, other):
+        try:
+            classof(self, other)
+        except TypeError:
+            return False
+        return (
+            self.shape == other.shape and list(self) == list(other))
+
+    def __len__(self):
+        return self.rows*self.cols
+
+    def __repr__(self):
+        return "_MinimalMatrix({}, {}, {})".format(self.rows, self.cols,
+                                                   self.mat)
+
+    @property
+    def shape(self):
+        return (self.rows, self.cols)
+
+
+class _CastableMatrix: # this is needed here ONLY FOR TESTS.
+    def as_mutable(self):
+        return self
+
+    def as_immutable(self):
+        return self
+
+
+class _MatrixWrapper:
+    """Wrapper class providing the minimum functionality for a matrix-like
+    object: .rows, .cols, .shape, indexability, and iterability. CommonMatrix
+    math operations should work on matrix-like objects. This one is intended for
+    matrix-like objects which use the same indexing format as SymPy with respect
+    to returning matrix elements instead of rows for non-tuple indexes.
+    """
+
+    is_Matrix     = False # needs to be here because of __getattr__
+    is_MatrixLike = True
+
+    def __init__(self, mat, shape):
+        self.mat = mat
+        self.shape = shape
+        self.rows, self.cols = shape
+
+    def __getitem__(self, key):
+        if isinstance(key, tuple):
+            return sympify(self.mat.__getitem__(key))
+
+        return sympify(self.mat.__getitem__((key // self.rows, key % self.cols)))
+
+    def __iter__(self): # supports numpy.matrix and numpy.array
+        mat = self.mat
+        cols = self.cols
+
+        return iter(sympify(mat[r, c]) for r in range(self.rows) for c in range(cols))
+
+
+class MatrixKind(Kind):
+    """
+    Kind for all matrices in SymPy.
+
+    Basic class for this kind is ``MatrixBase`` and ``MatrixExpr``,
+    but any expression representing the matrix can have this.
+
+    Parameters
+    ==========
+
+    element_kind : Kind
+        Kind of the element. Default is
+        :class:`sympy.core.kind.NumberKind`,
+        which means that the matrix contains only numbers.
+
+    Examples
+    ========
+
+    Any instance of matrix class has ``MatrixKind``:
+
+    >>> from sympy import MatrixSymbol
+    >>> A = MatrixSymbol('A', 2,2)
+    >>> A.kind
+    MatrixKind(NumberKind)
+
+    Although expression representing a matrix may be not instance of
+    matrix class, it will have ``MatrixKind`` as well:
+
+    >>> from sympy import MatrixExpr, Integral
+    >>> from sympy.abc import x
+    >>> intM = Integral(A, x)
+    >>> isinstance(intM, MatrixExpr)
+    False
+    >>> intM.kind
+    MatrixKind(NumberKind)
+
+    Use ``isinstance()`` to check for ``MatrixKind`` without specifying
+    the element kind. Use ``is`` with specifying the element kind:
+
+    >>> from sympy import Matrix
+    >>> from sympy.core import NumberKind
+    >>> from sympy.matrices import MatrixKind
+    >>> M = Matrix([1, 2])
+    >>> isinstance(M.kind, MatrixKind)
+    True
+    >>> M.kind is MatrixKind(NumberKind)
+    True
+
+    See Also
+    ========
+
+    sympy.core.kind.NumberKind
+    sympy.core.kind.UndefinedKind
+    sympy.core.containers.TupleKind
+    sympy.sets.sets.SetKind
+
+    """
+    def __new__(cls, element_kind=NumberKind):
+        obj = super().__new__(cls, element_kind)
+        obj.element_kind = element_kind
+        return obj
+
+    def __repr__(self):
+        return "MatrixKind(%s)" % self.element_kind
+
+
+def _matrixify(mat):
+    """If `mat` is a Matrix or is matrix-like,
+    return a Matrix or MatrixWrapper object.  Otherwise
+    `mat` is passed through without modification."""
+
+    if getattr(mat, 'is_Matrix', False) or getattr(mat, 'is_MatrixLike', False):
+        return mat
+
+    if not(getattr(mat, 'is_Matrix', True) or getattr(mat, 'is_MatrixLike', True)):
+        return mat
+
+    shape = None
+
+    if hasattr(mat, 'shape'): # numpy, scipy.sparse
+        if len(mat.shape) == 2:
+            shape = mat.shape
+    elif hasattr(mat, 'rows') and hasattr(mat, 'cols'): # mpmath
+        shape = (mat.rows, mat.cols)
+
+    if shape:
+        return _MatrixWrapper(mat, shape)
+
+    return mat
+
+
+def a2idx(j, n=None):
+    """Return integer after making positive and validating against n."""
+    if not isinstance(j, int):
+        jindex = getattr(j, '__index__', None)
+        if jindex is not None:
+            j = jindex()
+        else:
+            raise IndexError("Invalid index a[%r]" % (j,))
+    if n is not None:
+        if j < 0:
+            j += n
+        if not (j >= 0 and j < n):
+            raise IndexError("Index out of range: a[%s]" % (j,))
+    return int(j)
+
+
+def classof(A, B):
+    """
+    Get the type of the result when combining matrices of different types.
+
+    Currently the strategy is that immutability is contagious.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, ImmutableMatrix
+    >>> from sympy.matrices.common import classof
+    >>> M = Matrix([[1, 2], [3, 4]]) # a Mutable Matrix
+    >>> IM = ImmutableMatrix([[1, 2], [3, 4]])
+    >>> classof(M, IM)
+    <class 'sympy.matrices.immutable.ImmutableDenseMatrix'>
+    """
+    priority_A = getattr(A, '_class_priority', None)
+    priority_B = getattr(B, '_class_priority', None)
+    if None not in (priority_A, priority_B):
+        if A._class_priority > B._class_priority:
+            return A.__class__
+        else:
+            return B.__class__
+
+    try:
+        import numpy
+    except ImportError:
+        pass
+    else:
+        if isinstance(A, numpy.ndarray):
+            return B.__class__
+        if isinstance(B, numpy.ndarray):
+            return A.__class__
+
+    raise TypeError("Incompatible classes %s, %s" % (A.__class__, B.__class__))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/decompositions.py

@@ -0,0 +1,1629 @@
+import copy
+
+from sympy.core import S
+from sympy.core.function import expand_mul
+from sympy.functions.elementary.miscellaneous import Min, sqrt
+from sympy.functions.elementary.complexes import sign
+
+from .common import NonSquareMatrixError, NonPositiveDefiniteMatrixError
+from .utilities import _get_intermediate_simp, _iszero
+from .determinant import _find_reasonable_pivot_naive
+
+
+def _rank_decomposition(M, iszerofunc=_iszero, simplify=False):
+    r"""Returns a pair of matrices (`C`, `F`) with matching rank
+    such that `A = C F`.
+
+    Parameters
+    ==========
+
+    iszerofunc : Function, optional
+        A function used for detecting whether an element can
+        act as a pivot.  ``lambda x: x.is_zero`` is used by default.
+
+    simplify : Bool or Function, optional
+        A function used to simplify elements when looking for a
+        pivot. By default SymPy's ``simplify`` is used.
+
+    Returns
+    =======
+
+    (C, F) : Matrices
+        `C` and `F` are full-rank matrices with rank as same as `A`,
+        whose product gives `A`.
+
+        See Notes for additional mathematical details.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([
+    ...     [1, 3, 1, 4],
+    ...     [2, 7, 3, 9],
+    ...     [1, 5, 3, 1],
+    ...     [1, 2, 0, 8]
+    ... ])
+    >>> C, F = A.rank_decomposition()
+    >>> C
+    Matrix([
+    [1, 3, 4],
+    [2, 7, 9],
+    [1, 5, 1],
+    [1, 2, 8]])
+    >>> F
+    Matrix([
+    [1, 0, -2, 0],
+    [0, 1,  1, 0],
+    [0, 0,  0, 1]])
+    >>> C * F == A
+    True
+
+    Notes
+    =====
+
+    Obtaining `F`, an RREF of `A`, is equivalent to creating a
+    product
+
+    .. math::
+        E_n E_{n-1} ... E_1 A = F
+
+    where `E_n, E_{n-1}, \dots, E_1` are the elimination matrices or
+    permutation matrices equivalent to each row-reduction step.
+
+    The inverse of the same product of elimination matrices gives
+    `C`:
+
+    .. math::
+        C = \left(E_n E_{n-1} \dots E_1\right)^{-1}
+
+    It is not necessary, however, to actually compute the inverse:
+    the columns of `C` are those from the original matrix with the
+    same column indices as the indices of the pivot columns of `F`.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Rank_factorization
+
+    .. [2] Piziak, R.; Odell, P. L. (1 June 1999).
+        "Full Rank Factorization of Matrices".
+        Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882
+
+    See Also
+    ========
+
+    sympy.matrices.matrices.MatrixReductions.rref
+    """
+
+    F, pivot_cols = M.rref(simplify=simplify, iszerofunc=iszerofunc,
+            pivots=True)
+    rank = len(pivot_cols)
+
+    C = M.extract(range(M.rows), pivot_cols)
+    F = F[:rank, :]
+
+    return C, F
+
+
+def _liupc(M):
+    """Liu's algorithm, for pre-determination of the Elimination Tree of
+    the given matrix, used in row-based symbolic Cholesky factorization.
+
+    Examples
+    ========
+
+    >>> from sympy import SparseMatrix
+    >>> S = SparseMatrix([
+    ... [1, 0, 3, 2],
+    ... [0, 0, 1, 0],
+    ... [4, 0, 0, 5],
+    ... [0, 6, 7, 0]])
+    >>> S.liupc()
+    ([[0], [], [0], [1, 2]], [4, 3, 4, 4])
+
+    References
+    ==========
+
+    .. [1] Symbolic Sparse Cholesky Factorization using Elimination Trees,
+           Jeroen Van Grondelle (1999)
+           https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582
+    """
+    # Algorithm 2.4, p 17 of reference
+
+    # get the indices of the elements that are non-zero on or below diag
+    R = [[] for r in range(M.rows)]
+
+    for r, c, _ in M.row_list():
+        if c <= r:
+            R[r].append(c)
+
+    inf     = len(R)  # nothing will be this large
+    parent  = [inf]*M.rows
+    virtual = [inf]*M.rows
+
+    for r in range(M.rows):
+        for c in R[r][:-1]:
+            while virtual[c] < r:
+                t          = virtual[c]
+                virtual[c] = r
+                c          = t
+
+            if virtual[c] == inf:
+                parent[c] = virtual[c] = r
+
+    return R, parent
+
+def _row_structure_symbolic_cholesky(M):
+    """Symbolic cholesky factorization, for pre-determination of the
+    non-zero structure of the Cholesky factororization.
+
+    Examples
+    ========
+
+    >>> from sympy import SparseMatrix
+    >>> S = SparseMatrix([
+    ... [1, 0, 3, 2],
+    ... [0, 0, 1, 0],
+    ... [4, 0, 0, 5],
+    ... [0, 6, 7, 0]])
+    >>> S.row_structure_symbolic_cholesky()
+    [[0], [], [0], [1, 2]]
+
+    References
+    ==========
+
+    .. [1] Symbolic Sparse Cholesky Factorization using Elimination Trees,
+           Jeroen Van Grondelle (1999)
+           https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582
+    """
+
+    R, parent = M.liupc()
+    inf       = len(R)  # this acts as infinity
+    Lrow      = copy.deepcopy(R)
+
+    for k in range(M.rows):
+        for j in R[k]:
+            while j != inf and j != k:
+                Lrow[k].append(j)
+                j = parent[j]
+
+        Lrow[k] = sorted(set(Lrow[k]))
+
+    return Lrow
+
+
+def _cholesky(M, hermitian=True):
+    """Returns the Cholesky-type decomposition L of a matrix A
+    such that L * L.H == A if hermitian flag is True,
+    or L * L.T == A if hermitian is False.
+
+    A must be a Hermitian positive-definite matrix if hermitian is True,
+    or a symmetric matrix if it is False.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
+    >>> A.cholesky()
+    Matrix([
+    [ 5, 0, 0],
+    [ 3, 3, 0],
+    [-1, 1, 3]])
+    >>> A.cholesky() * A.cholesky().T
+    Matrix([
+    [25, 15, -5],
+    [15, 18,  0],
+    [-5,  0, 11]])
+
+    The matrix can have complex entries:
+
+    >>> from sympy import I
+    >>> A = Matrix(((9, 3*I), (-3*I, 5)))
+    >>> A.cholesky()
+    Matrix([
+    [ 3, 0],
+    [-I, 2]])
+    >>> A.cholesky() * A.cholesky().H
+    Matrix([
+    [   9, 3*I],
+    [-3*I,   5]])
+
+    Non-hermitian Cholesky-type decomposition may be useful when the
+    matrix is not positive-definite.
+
+    >>> A = Matrix([[1, 2], [2, 1]])
+    >>> L = A.cholesky(hermitian=False)
+    >>> L
+    Matrix([
+    [1,         0],
+    [2, sqrt(3)*I]])
+    >>> L*L.T == A
+    True
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.LDLdecomposition
+    sympy.matrices.matrices.MatrixBase.LUdecomposition
+    QRdecomposition
+    """
+
+    from .dense import MutableDenseMatrix
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if hermitian and not M.is_hermitian:
+        raise ValueError("Matrix must be Hermitian.")
+    if not hermitian and not M.is_symmetric():
+        raise ValueError("Matrix must be symmetric.")
+
+    L   = MutableDenseMatrix.zeros(M.rows, M.rows)
+
+    if hermitian:
+        for i in range(M.rows):
+            for j in range(i):
+                L[i, j] = ((1 / L[j, j])*(M[i, j] -
+                    sum(L[i, k]*L[j, k].conjugate() for k in range(j))))
+
+            Lii2 = (M[i, i] -
+                sum(L[i, k]*L[i, k].conjugate() for k in range(i)))
+
+            if Lii2.is_positive is False:
+                raise NonPositiveDefiniteMatrixError(
+                    "Matrix must be positive-definite")
+
+            L[i, i] = sqrt(Lii2)
+
+    else:
+        for i in range(M.rows):
+            for j in range(i):
+                L[i, j] = ((1 / L[j, j])*(M[i, j] -
+                    sum(L[i, k]*L[j, k] for k in range(j))))
+
+            L[i, i] = sqrt(M[i, i] -
+                sum(L[i, k]**2 for k in range(i)))
+
+    return M._new(L)
+
+def _cholesky_sparse(M, hermitian=True):
+    """
+    Returns the Cholesky decomposition L of a matrix A
+    such that L * L.T = A
+
+    A must be a square, symmetric, positive-definite
+    and non-singular matrix
+
+    Examples
+    ========
+
+    >>> from sympy import SparseMatrix
+    >>> A = SparseMatrix(((25,15,-5),(15,18,0),(-5,0,11)))
+    >>> A.cholesky()
+    Matrix([
+    [ 5, 0, 0],
+    [ 3, 3, 0],
+    [-1, 1, 3]])
+    >>> A.cholesky() * A.cholesky().T == A
+    True
+
+    The matrix can have complex entries:
+
+    >>> from sympy import I
+    >>> A = SparseMatrix(((9, 3*I), (-3*I, 5)))
+    >>> A.cholesky()
+    Matrix([
+    [ 3, 0],
+    [-I, 2]])
+    >>> A.cholesky() * A.cholesky().H
+    Matrix([
+    [   9, 3*I],
+    [-3*I,   5]])
+
+    Non-hermitian Cholesky-type decomposition may be useful when the
+    matrix is not positive-definite.
+
+    >>> A = SparseMatrix([[1, 2], [2, 1]])
+    >>> L = A.cholesky(hermitian=False)
+    >>> L
+    Matrix([
+    [1,         0],
+    [2, sqrt(3)*I]])
+    >>> L*L.T == A
+    True
+
+    See Also
+    ========
+
+    sympy.matrices.sparse.SparseMatrix.LDLdecomposition
+    sympy.matrices.matrices.MatrixBase.LUdecomposition
+    QRdecomposition
+    """
+
+    from .dense import MutableDenseMatrix
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if hermitian and not M.is_hermitian:
+        raise ValueError("Matrix must be Hermitian.")
+    if not hermitian and not M.is_symmetric():
+        raise ValueError("Matrix must be symmetric.")
+
+    dps       = _get_intermediate_simp(expand_mul, expand_mul)
+    Crowstruc = M.row_structure_symbolic_cholesky()
+    C         = MutableDenseMatrix.zeros(M.rows)
+
+    for i in range(len(Crowstruc)):
+        for j in Crowstruc[i]:
+            if i != j:
+                C[i, j] = M[i, j]
+                summ    = 0
+
+                for p1 in Crowstruc[i]:
+                    if p1 < j:
+                        for p2 in Crowstruc[j]:
+                            if p2 < j:
+                                if p1 == p2:
+                                    if hermitian:
+                                        summ += C[i, p1]*C[j, p1].conjugate()
+                                    else:
+                                        summ += C[i, p1]*C[j, p1]
+                            else:
+                                break
+                        else:
+                            break
+
+                C[i, j] = dps((C[i, j] - summ) / C[j, j])
+
+            else: # i == j
+                C[j, j] = M[j, j]
+                summ    = 0
+
+                for k in Crowstruc[j]:
+                    if k < j:
+                        if hermitian:
+                            summ += C[j, k]*C[j, k].conjugate()
+                        else:
+                            summ += C[j, k]**2
+                    else:
+                        break
+
+                Cjj2 = dps(C[j, j] - summ)
+
+                if hermitian and Cjj2.is_positive is False:
+                    raise NonPositiveDefiniteMatrixError(
+                        "Matrix must be positive-definite")
+
+                C[j, j] = sqrt(Cjj2)
+
+    return M._new(C)
+
+
+def _LDLdecomposition(M, hermitian=True):
+    """Returns the LDL Decomposition (L, D) of matrix A,
+    such that L * D * L.H == A if hermitian flag is True, or
+    L * D * L.T == A if hermitian is False.
+    This method eliminates the use of square root.
+    Further this ensures that all the diagonal entries of L are 1.
+    A must be a Hermitian positive-definite matrix if hermitian is True,
+    or a symmetric matrix otherwise.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, eye
+    >>> A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
+    >>> L, D = A.LDLdecomposition()
+    >>> L
+    Matrix([
+    [   1,   0, 0],
+    [ 3/5,   1, 0],
+    [-1/5, 1/3, 1]])
+    >>> D
+    Matrix([
+    [25, 0, 0],
+    [ 0, 9, 0],
+    [ 0, 0, 9]])
+    >>> L * D * L.T * A.inv() == eye(A.rows)
+    True
+
+    The matrix can have complex entries:
+
+    >>> from sympy import I
+    >>> A = Matrix(((9, 3*I), (-3*I, 5)))
+    >>> L, D = A.LDLdecomposition()
+    >>> L
+    Matrix([
+    [   1, 0],
+    [-I/3, 1]])
+    >>> D
+    Matrix([
+    [9, 0],
+    [0, 4]])
+    >>> L*D*L.H == A
+    True
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.cholesky
+    sympy.matrices.matrices.MatrixBase.LUdecomposition
+    QRdecomposition
+    """
+
+    from .dense import MutableDenseMatrix
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if hermitian and not M.is_hermitian:
+        raise ValueError("Matrix must be Hermitian.")
+    if not hermitian and not M.is_symmetric():
+        raise ValueError("Matrix must be symmetric.")
+
+    D   = MutableDenseMatrix.zeros(M.rows, M.rows)
+    L   = MutableDenseMatrix.eye(M.rows)
+
+    if hermitian:
+        for i in range(M.rows):
+            for j in range(i):
+                L[i, j] = (1 / D[j, j])*(M[i, j] - sum(
+                    L[i, k]*L[j, k].conjugate()*D[k, k] for k in range(j)))
+
+            D[i, i] = (M[i, i] -
+                sum(L[i, k]*L[i, k].conjugate()*D[k, k] for k in range(i)))
+
+            if D[i, i].is_positive is False:
+                raise NonPositiveDefiniteMatrixError(
+                    "Matrix must be positive-definite")
+
+    else:
+        for i in range(M.rows):
+            for j in range(i):
+                L[i, j] = (1 / D[j, j])*(M[i, j] - sum(
+                    L[i, k]*L[j, k]*D[k, k] for k in range(j)))
+
+            D[i, i] = M[i, i] - sum(L[i, k]**2*D[k, k] for k in range(i))
+
+    return M._new(L), M._new(D)
+
+def _LDLdecomposition_sparse(M, hermitian=True):
+    """
+    Returns the LDL Decomposition (matrices ``L`` and ``D``) of matrix
+    ``A``, such that ``L * D * L.T == A``. ``A`` must be a square,
+    symmetric, positive-definite and non-singular.
+
+    This method eliminates the use of square root and ensures that all
+    the diagonal entries of L are 1.
+
+    Examples
+    ========
+
+    >>> from sympy import SparseMatrix
+    >>> A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
+    >>> L, D = A.LDLdecomposition()
+    >>> L
+    Matrix([
+    [   1,   0, 0],
+    [ 3/5,   1, 0],
+    [-1/5, 1/3, 1]])
+    >>> D
+    Matrix([
+    [25, 0, 0],
+    [ 0, 9, 0],
+    [ 0, 0, 9]])
+    >>> L * D * L.T == A
+    True
+
+    """
+
+    from .dense import MutableDenseMatrix
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+    if hermitian and not M.is_hermitian:
+        raise ValueError("Matrix must be Hermitian.")
+    if not hermitian and not M.is_symmetric():
+        raise ValueError("Matrix must be symmetric.")
+
+    dps       = _get_intermediate_simp(expand_mul, expand_mul)
+    Lrowstruc = M.row_structure_symbolic_cholesky()
+    L         = MutableDenseMatrix.eye(M.rows)
+    D         = MutableDenseMatrix.zeros(M.rows, M.cols)
+
+    for i in range(len(Lrowstruc)):
+        for j in Lrowstruc[i]:
+            if i != j:
+                L[i, j] = M[i, j]
+                summ    = 0
+
+                for p1 in Lrowstruc[i]:
+                    if p1 < j:
+                        for p2 in Lrowstruc[j]:
+                            if p2 < j:
+                                if p1 == p2:
+                                    if hermitian:
+                                        summ += L[i, p1]*L[j, p1].conjugate()*D[p1, p1]
+                                    else:
+                                        summ += L[i, p1]*L[j, p1]*D[p1, p1]
+                            else:
+                                break
+                    else:
+                        break
+
+                L[i, j] = dps((L[i, j] - summ) / D[j, j])
+
+            else: # i == j
+                D[i, i] = M[i, i]
+                summ    = 0
+
+                for k in Lrowstruc[i]:
+                    if k < i:
+                        if hermitian:
+                            summ += L[i, k]*L[i, k].conjugate()*D[k, k]
+                        else:
+                            summ += L[i, k]**2*D[k, k]
+                    else:
+                        break
+
+                D[i, i] = dps(D[i, i] - summ)
+
+                if hermitian and D[i, i].is_positive is False:
+                    raise NonPositiveDefiniteMatrixError(
+                        "Matrix must be positive-definite")
+
+    return M._new(L), M._new(D)
+
+
+def _LUdecomposition(M, iszerofunc=_iszero, simpfunc=None, rankcheck=False):
+    """Returns (L, U, perm) where L is a lower triangular matrix with unit
+    diagonal, U is an upper triangular matrix, and perm is a list of row
+    swap index pairs. If A is the original matrix, then
+    ``A = (L*U).permuteBkwd(perm)``, and the row permutation matrix P such
+    that $P A = L U$ can be computed by ``P = eye(A.rows).permuteFwd(perm)``.
+
+    See documentation for LUCombined for details about the keyword argument
+    rankcheck, iszerofunc, and simpfunc.
+
+    Parameters
+    ==========
+
+    rankcheck : bool, optional
+        Determines if this function should detect the rank
+        deficiency of the matrixis and should raise a
+        ``ValueError``.
+
+    iszerofunc : function, optional
+        A function which determines if a given expression is zero.
+
+        The function should be a callable that takes a single
+        SymPy expression and returns a 3-valued boolean value
+        ``True``, ``False``, or ``None``.
+
+        It is internally used by the pivot searching algorithm.
+        See the notes section for a more information about the
+        pivot searching algorithm.
+
+    simpfunc : function or None, optional
+        A function that simplifies the input.
+
+        If this is specified as a function, this function should be
+        a callable that takes a single SymPy expression and returns
+        an another SymPy expression that is algebraically
+        equivalent.
+
+        If ``None``, it indicates that the pivot search algorithm
+        should not attempt to simplify any candidate pivots.
+
+        It is internally used by the pivot searching algorithm.
+        See the notes section for a more information about the
+        pivot searching algorithm.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> a = Matrix([[4, 3], [6, 3]])
+    >>> L, U, _ = a.LUdecomposition()
+    >>> L
+    Matrix([
+    [  1, 0],
+    [3/2, 1]])
+    >>> U
+    Matrix([
+    [4,    3],
+    [0, -3/2]])
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.cholesky
+    sympy.matrices.dense.DenseMatrix.LDLdecomposition
+    QRdecomposition
+    LUdecomposition_Simple
+    LUdecompositionFF
+    LUsolve
+    """
+
+    combined, p = M.LUdecomposition_Simple(iszerofunc=iszerofunc,
+        simpfunc=simpfunc, rankcheck=rankcheck)
+
+    # L is lower triangular ``M.rows x M.rows``
+    # U is upper triangular ``M.rows x M.cols``
+    # L has unit diagonal. For each column in combined, the subcolumn
+    # below the diagonal of combined is shared by L.
+    # If L has more columns than combined, then the remaining subcolumns
+    # below the diagonal of L are zero.
+    # The upper triangular portion of L and combined are equal.
+    def entry_L(i, j):
+        if i < j:
+            # Super diagonal entry
+            return M.zero
+        elif i == j:
+            return M.one
+        elif j < combined.cols:
+            return combined[i, j]
+
+        # Subdiagonal entry of L with no corresponding
+        # entry in combined
+        return M.zero
+
+    def entry_U(i, j):
+        return M.zero if i > j else combined[i, j]
+
+    L = M._new(combined.rows, combined.rows, entry_L)
+    U = M._new(combined.rows, combined.cols, entry_U)
+
+    return L, U, p
+
+def _LUdecomposition_Simple(M, iszerofunc=_iszero, simpfunc=None,
+        rankcheck=False):
+    r"""Compute the PLU decomposition of the matrix.
+
+    Parameters
+    ==========
+
+    rankcheck : bool, optional
+        Determines if this function should detect the rank
+        deficiency of the matrixis and should raise a
+        ``ValueError``.
+
+    iszerofunc : function, optional
+        A function which determines if a given expression is zero.
+
+        The function should be a callable that takes a single
+        SymPy expression and returns a 3-valued boolean value
+        ``True``, ``False``, or ``None``.
+
+        It is internally used by the pivot searching algorithm.
+        See the notes section for a more information about the
+        pivot searching algorithm.
+
+    simpfunc : function or None, optional
+        A function that simplifies the input.
+
+        If this is specified as a function, this function should be
+        a callable that takes a single SymPy expression and returns
+        an another SymPy expression that is algebraically
+        equivalent.
+
+        If ``None``, it indicates that the pivot search algorithm
+        should not attempt to simplify any candidate pivots.
+
+        It is internally used by the pivot searching algorithm.
+        See the notes section for a more information about the
+        pivot searching algorithm.
+
+    Returns
+    =======
+
+    (lu, row_swaps) : (Matrix, list)
+        If the original matrix is a $m, n$ matrix:
+
+        *lu* is a $m, n$ matrix, which contains result of the
+        decomposition in a compressed form. See the notes section
+        to see how the matrix is compressed.
+
+        *row_swaps* is a $m$-element list where each element is a
+        pair of row exchange indices.
+
+        ``A = (L*U).permute_backward(perm)``, and the row
+        permutation matrix $P$ from the formula $P A = L U$ can be
+        computed by ``P=eye(A.row).permute_forward(perm)``.
+
+    Raises
+    ======
+
+    ValueError
+        Raised if ``rankcheck=True`` and the matrix is found to
+        be rank deficient during the computation.
+
+    Notes
+    =====
+
+    About the PLU decomposition:
+
+    PLU decomposition is a generalization of a LU decomposition
+    which can be extended for rank-deficient matrices.
+
+    It can further be generalized for non-square matrices, and this
+    is the notation that SymPy is using.
+
+    PLU decomposition is a decomposition of a $m, n$ matrix $A$ in
+    the form of $P A = L U$ where
+
+    * $L$ is a $m, m$ lower triangular matrix with unit diagonal
+        entries.
+    * $U$ is a $m, n$ upper triangular matrix.
+    * $P$ is a $m, m$ permutation matrix.
+
+    So, for a square matrix, the decomposition would look like:
+
+    .. math::
+        L = \begin{bmatrix}
+        1 & 0 & 0 & \cdots & 0 \\
+        L_{1, 0} & 1 & 0 & \cdots & 0 \\
+        L_{2, 0} & L_{2, 1} & 1 & \cdots & 0 \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        L_{n-1, 0} & L_{n-1, 1} & L_{n-1, 2} & \cdots & 1
+        \end{bmatrix}
+
+    .. math::
+        U = \begin{bmatrix}
+        U_{0, 0} & U_{0, 1} & U_{0, 2} & \cdots & U_{0, n-1} \\
+        0 & U_{1, 1} & U_{1, 2} & \cdots & U_{1, n-1} \\
+        0 & 0 & U_{2, 2} & \cdots & U_{2, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        0 & 0 & 0 & \cdots & U_{n-1, n-1}
+        \end{bmatrix}
+
+    And for a matrix with more rows than the columns,
+    the decomposition would look like:
+
+    .. math::
+        L = \begin{bmatrix}
+        1 & 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
+        L_{1, 0} & 1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\
+        L_{2, 0} & L_{2, 1} & 1 & \cdots & 0 & 0 & \cdots & 0 \\
+        \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots
+        & \vdots \\
+        L_{n-1, 0} & L_{n-1, 1} & L_{n-1, 2} & \cdots & 1 & 0
+        & \cdots & 0 \\
+        L_{n, 0} & L_{n, 1} & L_{n, 2} & \cdots & L_{n, n-1} & 1
+        & \cdots & 0 \\
+        \vdots & \vdots & \vdots & \ddots & \vdots & \vdots
+        & \ddots & \vdots \\
+        L_{m-1, 0} & L_{m-1, 1} & L_{m-1, 2} & \cdots & L_{m-1, n-1}
+        & 0 & \cdots & 1 \\
+        \end{bmatrix}
+
+    .. math::
+        U = \begin{bmatrix}
+        U_{0, 0} & U_{0, 1} & U_{0, 2} & \cdots & U_{0, n-1} \\
+        0 & U_{1, 1} & U_{1, 2} & \cdots & U_{1, n-1} \\
+        0 & 0 & U_{2, 2} & \cdots & U_{2, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        0 & 0 & 0 & \cdots & U_{n-1, n-1} \\
+        0 & 0 & 0 & \cdots & 0 \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        0 & 0 & 0 & \cdots & 0
+        \end{bmatrix}
+
+    Finally, for a matrix with more columns than the rows, the
+    decomposition would look like:
+
+    .. math::
+        L = \begin{bmatrix}
+        1 & 0 & 0 & \cdots & 0 \\
+        L_{1, 0} & 1 & 0 & \cdots & 0 \\
+        L_{2, 0} & L_{2, 1} & 1 & \cdots & 0 \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        L_{m-1, 0} & L_{m-1, 1} & L_{m-1, 2} & \cdots & 1
+        \end{bmatrix}
+
+    .. math::
+        U = \begin{bmatrix}
+        U_{0, 0} & U_{0, 1} & U_{0, 2} & \cdots & U_{0, m-1}
+        & \cdots & U_{0, n-1} \\
+        0 & U_{1, 1} & U_{1, 2} & \cdots & U_{1, m-1}
+        & \cdots & U_{1, n-1} \\
+        0 & 0 & U_{2, 2} & \cdots & U_{2, m-1}
+        & \cdots & U_{2, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots
+        & \cdots & \vdots \\
+        0 & 0 & 0 & \cdots & U_{m-1, m-1}
+        & \cdots & U_{m-1, n-1} \\
+        \end{bmatrix}
+
+    About the compressed LU storage:
+
+    The results of the decomposition are often stored in compressed
+    forms rather than returning $L$ and $U$ matrices individually.
+
+    It may be less intiuitive, but it is commonly used for a lot of
+    numeric libraries because of the efficiency.
+
+    The storage matrix is defined as following for this specific
+    method:
+
+    * The subdiagonal elements of $L$ are stored in the subdiagonal
+        portion of $LU$, that is $LU_{i, j} = L_{i, j}$ whenever
+        $i > j$.
+    * The elements on the diagonal of $L$ are all 1, and are not
+        explicitly stored.
+    * $U$ is stored in the upper triangular portion of $LU$, that is
+        $LU_{i, j} = U_{i, j}$ whenever $i <= j$.
+    * For a case of $m > n$, the right side of the $L$ matrix is
+        trivial to store.
+    * For a case of $m < n$, the below side of the $U$ matrix is
+        trivial to store.
+
+    So, for a square matrix, the compressed output matrix would be:
+
+    .. math::
+        LU = \begin{bmatrix}
+        U_{0, 0} & U_{0, 1} & U_{0, 2} & \cdots & U_{0, n-1} \\
+        L_{1, 0} & U_{1, 1} & U_{1, 2} & \cdots & U_{1, n-1} \\
+        L_{2, 0} & L_{2, 1} & U_{2, 2} & \cdots & U_{2, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        L_{n-1, 0} & L_{n-1, 1} & L_{n-1, 2} & \cdots & U_{n-1, n-1}
+        \end{bmatrix}
+
+    For a matrix with more rows than the columns, the compressed
+    output matrix would be:
+
+    .. math::
+        LU = \begin{bmatrix}
+        U_{0, 0} & U_{0, 1} & U_{0, 2} & \cdots & U_{0, n-1} \\
+        L_{1, 0} & U_{1, 1} & U_{1, 2} & \cdots & U_{1, n-1} \\
+        L_{2, 0} & L_{2, 1} & U_{2, 2} & \cdots & U_{2, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        L_{n-1, 0} & L_{n-1, 1} & L_{n-1, 2} & \cdots
+        & U_{n-1, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots \\
+        L_{m-1, 0} & L_{m-1, 1} & L_{m-1, 2} & \cdots
+        & L_{m-1, n-1} \\
+        \end{bmatrix}
+
+    For a matrix with more columns than the rows, the compressed
+    output matrix would be:
+
+    .. math::
+        LU = \begin{bmatrix}
+        U_{0, 0} & U_{0, 1} & U_{0, 2} & \cdots & U_{0, m-1}
+        & \cdots & U_{0, n-1} \\
+        L_{1, 0} & U_{1, 1} & U_{1, 2} & \cdots & U_{1, m-1}
+        & \cdots & U_{1, n-1} \\
+        L_{2, 0} & L_{2, 1} & U_{2, 2} & \cdots & U_{2, m-1}
+        & \cdots & U_{2, n-1} \\
+        \vdots & \vdots & \vdots & \ddots & \vdots
+        & \cdots & \vdots \\
+        L_{m-1, 0} & L_{m-1, 1} & L_{m-1, 2} & \cdots & U_{m-1, m-1}
+        & \cdots & U_{m-1, n-1} \\
+        \end{bmatrix}
+
+    About the pivot searching algorithm:
+
+    When a matrix contains symbolic entries, the pivot search algorithm
+    differs from the case where every entry can be categorized as zero or
+    nonzero.
+    The algorithm searches column by column through the submatrix whose
+    top left entry coincides with the pivot position.
+    If it exists, the pivot is the first entry in the current search
+    column that iszerofunc guarantees is nonzero.
+    If no such candidate exists, then each candidate pivot is simplified
+    if simpfunc is not None.
+    The search is repeated, with the difference that a candidate may be
+    the pivot if ``iszerofunc()`` cannot guarantee that it is nonzero.
+    In the second search the pivot is the first candidate that
+    iszerofunc can guarantee is nonzero.
+    If no such candidate exists, then the pivot is the first candidate
+    for which iszerofunc returns None.
+    If no such candidate exists, then the search is repeated in the next
+    column to the right.
+    The pivot search algorithm differs from the one in ``rref()``, which
+    relies on ``_find_reasonable_pivot()``.
+    Future versions of ``LUdecomposition_simple()`` may use
+    ``_find_reasonable_pivot()``.
+
+    See Also
+    ========
+
+    sympy.matrices.matrices.MatrixBase.LUdecomposition
+    LUdecompositionFF
+    LUsolve
+    """
+
+    if rankcheck:
+        # https://github.com/sympy/sympy/issues/9796
+        pass
+
+    if S.Zero in M.shape:
+        # Define LU decomposition of a matrix with no entries as a matrix
+        # of the same dimensions with all zero entries.
+        return M.zeros(M.rows, M.cols), []
+
+    dps       = _get_intermediate_simp()
+    lu        = M.as_mutable()
+    row_swaps = []
+
+    pivot_col = 0
+
+    for pivot_row in range(0, lu.rows - 1):
+        # Search for pivot. Prefer entry that iszeropivot determines
+        # is nonzero, over entry that iszeropivot cannot guarantee
+        # is  zero.
+        # XXX ``_find_reasonable_pivot`` uses slow zero testing. Blocked by bug #10279
+        # Future versions of LUdecomposition_simple can pass iszerofunc and simpfunc
+        # to _find_reasonable_pivot().
+        # In pass 3 of _find_reasonable_pivot(), the predicate in ``if x.equals(S.Zero):``
+        # calls sympy.simplify(), and not the simplification function passed in via
+        # the keyword argument simpfunc.
+        iszeropivot = True
+
+        while pivot_col != M.cols and iszeropivot:
+            sub_col = (lu[r, pivot_col] for r in range(pivot_row, M.rows))
+
+            pivot_row_offset, pivot_value, is_assumed_non_zero, ind_simplified_pairs =\
+                _find_reasonable_pivot_naive(sub_col, iszerofunc, simpfunc)
+
+            iszeropivot = pivot_value is None
+
+            if iszeropivot:
+                # All candidate pivots in this column are zero.
+                # Proceed to next column.
+                pivot_col += 1
+
+        if rankcheck and pivot_col != pivot_row:
+            # All entries including and below the pivot position are
+            # zero, which indicates that the rank of the matrix is
+            # strictly less than min(num rows, num cols)
+            # Mimic behavior of previous implementation, by throwing a
+            # ValueError.
+            raise ValueError("Rank of matrix is strictly less than"
+                                " number of rows or columns."
+                                " Pass keyword argument"
+                                " rankcheck=False to compute"
+                                " the LU decomposition of this matrix.")
+
+        candidate_pivot_row = None if pivot_row_offset is None else pivot_row + pivot_row_offset
+
+        if candidate_pivot_row is None and iszeropivot:
+            # If candidate_pivot_row is None and iszeropivot is True
+            # after pivot search has completed, then the submatrix
+            # below and to the right of (pivot_row, pivot_col) is
+            # all zeros, indicating that Gaussian elimination is
+            # complete.
+            return lu, row_swaps
+
+        # Update entries simplified during pivot search.
+        for offset, val in ind_simplified_pairs:
+            lu[pivot_row + offset, pivot_col] = val
+
+        if pivot_row != candidate_pivot_row:
+            # Row swap book keeping:
+            # Record which rows were swapped.
+            # Update stored portion of L factor by multiplying L on the
+            # left and right with the current permutation.
+            # Swap rows of U.
+            row_swaps.append([pivot_row, candidate_pivot_row])
+
+            # Update L.
+            lu[pivot_row, 0:pivot_row], lu[candidate_pivot_row, 0:pivot_row] = \
+                lu[candidate_pivot_row, 0:pivot_row], lu[pivot_row, 0:pivot_row]
+
+            # Swap pivot row of U with candidate pivot row.
+            lu[pivot_row, pivot_col:lu.cols], lu[candidate_pivot_row, pivot_col:lu.cols] = \
+                lu[candidate_pivot_row, pivot_col:lu.cols], lu[pivot_row, pivot_col:lu.cols]
+
+        # Introduce zeros below the pivot by adding a multiple of the
+        # pivot row to a row under it, and store the result in the
+        # row under it.
+        # Only entries in the target row whose index is greater than
+        # start_col may be nonzero.
+        start_col = pivot_col + 1
+
+        for row in range(pivot_row + 1, lu.rows):
+            # Store factors of L in the subcolumn below
+            # (pivot_row, pivot_row).
+            lu[row, pivot_row] = \
+                dps(lu[row, pivot_col]/lu[pivot_row, pivot_col])
+
+            # Form the linear combination of the pivot row and the current
+            # row below the pivot row that zeros the entries below the pivot.
+            # Employing slicing instead of a loop here raises
+            # NotImplementedError: Cannot add Zero to MutableSparseMatrix
+            # in sympy/matrices/tests/test_sparse.py.
+            # c = pivot_row + 1 if pivot_row == pivot_col else pivot_col
+            for c in range(start_col, lu.cols):
+                lu[row, c] = dps(lu[row, c] - lu[row, pivot_row]*lu[pivot_row, c])
+
+        if pivot_row != pivot_col:
+            # matrix rank < min(num rows, num cols),
+            # so factors of L are not stored directly below the pivot.
+            # These entries are zero by construction, so don't bother
+            # computing them.
+            for row in range(pivot_row + 1, lu.rows):
+                lu[row, pivot_col] = M.zero
+
+        pivot_col += 1
+
+        if pivot_col == lu.cols:
+            # All candidate pivots are zero implies that Gaussian
+            # elimination is complete.
+            return lu, row_swaps
+
+    if rankcheck:
+        if iszerofunc(
+                lu[Min(lu.rows, lu.cols) - 1, Min(lu.rows, lu.cols) - 1]):
+            raise ValueError("Rank of matrix is strictly less than"
+                                " number of rows or columns."
+                                " Pass keyword argument"
+                                " rankcheck=False to compute"
+                                " the LU decomposition of this matrix.")
+
+    return lu, row_swaps
+
+def _LUdecompositionFF(M):
+    """Compute a fraction-free LU decomposition.
+
+    Returns 4 matrices P, L, D, U such that PA = L D**-1 U.
+    If the elements of the matrix belong to some integral domain I, then all
+    elements of L, D and U are guaranteed to belong to I.
+
+    See Also
+    ========
+
+    sympy.matrices.matrices.MatrixBase.LUdecomposition
+    LUdecomposition_Simple
+    LUsolve
+
+    References
+    ==========
+
+    .. [1] W. Zhou & D.J. Jeffrey, "Fraction-free matrix factors: new forms
+        for LU and QR factors". Frontiers in Computer Science in China,
+        Vol 2, no. 1, pp. 67-80, 2008.
+    """
+
+    from sympy.matrices import SparseMatrix
+
+    zeros    = SparseMatrix.zeros
+    eye      = SparseMatrix.eye
+    n, m     = M.rows, M.cols
+    U, L, P  = M.as_mutable(), eye(n), eye(n)
+    DD       = zeros(n, n)
+    oldpivot = 1
+
+    for k in range(n - 1):
+        if U[k, k] == 0:
+            for kpivot in range(k + 1, n):
+                if U[kpivot, k]:
+                    break
+            else:
+                raise ValueError("Matrix is not full rank")
+
+            U[k, k:], U[kpivot, k:] = U[kpivot, k:], U[k, k:]
+            L[k, :k], L[kpivot, :k] = L[kpivot, :k], L[k, :k]
+            P[k, :], P[kpivot, :]   = P[kpivot, :], P[k, :]
+
+        L [k, k] = Ukk = U[k, k]
+        DD[k, k] = oldpivot * Ukk
+
+        for i in range(k + 1, n):
+            L[i, k] = Uik = U[i, k]
+
+            for j in range(k + 1, m):
+                U[i, j] = (Ukk * U[i, j] - U[k, j] * Uik) / oldpivot
+
+            U[i, k] = 0
+
+        oldpivot = Ukk
+
+    DD[n - 1, n - 1] = oldpivot
+
+    return P, L, DD, U
+
+def _singular_value_decomposition(A):
+    r"""Returns a Condensed Singular Value decomposition.
+
+    Explanation
+    ===========
+
+    A Singular Value decomposition is a decomposition in the form $A = U \Sigma V$
+    where
+
+    - $U, V$ are column orthogonal matrix.
+    - $\Sigma$ is a diagonal matrix, where the main diagonal contains singular
+      values of matrix A.
+
+    A column orthogonal matrix satisfies
+    $\mathbb{I} = U^H U$ while a full orthogonal matrix satisfies
+    relation $\mathbb{I} = U U^H = U^H U$ where $\mathbb{I}$ is an identity
+    matrix with matching dimensions.
+
+    For matrices which are not square or are rank-deficient, it is
+    sufficient to return a column orthogonal matrix because augmenting
+    them may introduce redundant computations.
+    In condensed Singular Value Decomposition we only return column orthogonal
+    matrices because of this reason
+
+    If you want to augment the results to return a full orthogonal
+    decomposition, you should use the following procedures.
+
+    - Augment the $U, V$ matrices with columns that are orthogonal to every
+      other columns and make it square.
+    - Augment the $\Sigma$ matrix with zero rows to make it have the same
+      shape as the original matrix.
+
+    The procedure will be illustrated in the examples section.
+
+    Examples
+    ========
+
+    we take a full rank matrix first:
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([[1, 2],[2,1]])
+    >>> U, S, V = A.singular_value_decomposition()
+    >>> U
+    Matrix([
+    [ sqrt(2)/2, sqrt(2)/2],
+    [-sqrt(2)/2, sqrt(2)/2]])
+    >>> S
+    Matrix([
+    [1, 0],
+    [0, 3]])
+    >>> V
+    Matrix([
+    [-sqrt(2)/2, sqrt(2)/2],
+    [ sqrt(2)/2, sqrt(2)/2]])
+
+    If a matrix if square and full rank both U, V
+    are orthogonal in both directions
+
+    >>> U * U.H
+    Matrix([
+    [1, 0],
+    [0, 1]])
+    >>> U.H * U
+    Matrix([
+    [1, 0],
+    [0, 1]])
+
+    >>> V * V.H
+    Matrix([
+    [1, 0],
+    [0, 1]])
+    >>> V.H * V
+    Matrix([
+    [1, 0],
+    [0, 1]])
+    >>> A == U * S * V.H
+    True
+
+    >>> C = Matrix([
+    ...         [1, 0, 0, 0, 2],
+    ...         [0, 0, 3, 0, 0],
+    ...         [0, 0, 0, 0, 0],
+    ...         [0, 2, 0, 0, 0],
+    ...     ])
+    >>> U, S, V = C.singular_value_decomposition()
+
+    >>> V.H * V
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+    >>> V * V.H
+    Matrix([
+    [1/5, 0, 0, 0, 2/5],
+    [  0, 1, 0, 0,   0],
+    [  0, 0, 1, 0,   0],
+    [  0, 0, 0, 0,   0],
+    [2/5, 0, 0, 0, 4/5]])
+
+    If you want to augment the results to be a full orthogonal
+    decomposition, you should augment $V$ with an another orthogonal
+    column.
+
+    You are able to append an arbitrary standard basis that are linearly
+    independent to every other columns and you can run the Gram-Schmidt
+    process to make them augmented as orthogonal basis.
+
+    >>> V_aug = V.row_join(Matrix([[0,0,0,0,1],
+    ... [0,0,0,1,0]]).H)
+    >>> V_aug = V_aug.QRdecomposition()[0]
+    >>> V_aug
+    Matrix([
+    [0,   sqrt(5)/5, 0, -2*sqrt(5)/5, 0],
+    [1,           0, 0,            0, 0],
+    [0,           0, 1,            0, 0],
+    [0,           0, 0,            0, 1],
+    [0, 2*sqrt(5)/5, 0,    sqrt(5)/5, 0]])
+    >>> V_aug.H * V_aug
+    Matrix([
+    [1, 0, 0, 0, 0],
+    [0, 1, 0, 0, 0],
+    [0, 0, 1, 0, 0],
+    [0, 0, 0, 1, 0],
+    [0, 0, 0, 0, 1]])
+    >>> V_aug * V_aug.H
+    Matrix([
+    [1, 0, 0, 0, 0],
+    [0, 1, 0, 0, 0],
+    [0, 0, 1, 0, 0],
+    [0, 0, 0, 1, 0],
+    [0, 0, 0, 0, 1]])
+
+    Similarly we augment U
+
+    >>> U_aug = U.row_join(Matrix([0,0,1,0]))
+    >>> U_aug = U_aug.QRdecomposition()[0]
+    >>> U_aug
+    Matrix([
+    [0, 1, 0, 0],
+    [0, 0, 1, 0],
+    [0, 0, 0, 1],
+    [1, 0, 0, 0]])
+
+    >>> U_aug.H * U_aug
+    Matrix([
+    [1, 0, 0, 0],
+    [0, 1, 0, 0],
+    [0, 0, 1, 0],
+    [0, 0, 0, 1]])
+    >>> U_aug * U_aug.H
+    Matrix([
+    [1, 0, 0, 0],
+    [0, 1, 0, 0],
+    [0, 0, 1, 0],
+    [0, 0, 0, 1]])
+
+    We add 2 zero columns and one row to S
+
+    >>> S_aug = S.col_join(Matrix([[0,0,0]]))
+    >>> S_aug = S_aug.row_join(Matrix([[0,0,0,0],
+    ... [0,0,0,0]]).H)
+    >>> S_aug
+    Matrix([
+    [2,       0, 0, 0, 0],
+    [0, sqrt(5), 0, 0, 0],
+    [0,       0, 3, 0, 0],
+    [0,       0, 0, 0, 0]])
+
+
+
+    >>> U_aug * S_aug * V_aug.H == C
+    True
+
+    """
+
+    AH = A.H
+    m, n = A.shape
+    if m >= n:
+        V, S = (AH * A).diagonalize()
+
+        ranked = []
+        for i, x in enumerate(S.diagonal()):
+            if not x.is_zero:
+                ranked.append(i)
+
+        V = V[:, ranked]
+
+        Singular_vals = [sqrt(S[i, i]) for i in range(S.rows) if i in ranked]
+
+        S = S.zeros(len(Singular_vals))
+
+        for i, sv in enumerate(Singular_vals):
+            S[i, i] = sv
+
+        V, _ = V.QRdecomposition()
+        U = A * V * S.inv()
+    else:
+        U, S = (A * AH).diagonalize()
+
+        ranked = []
+        for i, x in enumerate(S.diagonal()):
+            if not x.is_zero:
+                ranked.append(i)
+
+        U = U[:, ranked]
+        Singular_vals = [sqrt(S[i, i]) for i in range(S.rows) if i in ranked]
+
+        S = S.zeros(len(Singular_vals))
+
+        for i, sv in enumerate(Singular_vals):
+            S[i, i] = sv
+
+        U, _ = U.QRdecomposition()
+        V = AH * U * S.inv()
+
+    return U, S, V
+
+def _QRdecomposition_optional(M, normalize=True):
+    def dot(u, v):
+        return u.dot(v, hermitian=True)
+
+    dps = _get_intermediate_simp(expand_mul, expand_mul)
+
+    A = M.as_mutable()
+    ranked = []
+
+    Q = A
+    R = A.zeros(A.cols)
+
+    for j in range(A.cols):
+        for i in range(j):
+            if Q[:, i].is_zero_matrix:
+                continue
+
+            R[i, j] = dot(Q[:, i], Q[:, j]) / dot(Q[:, i], Q[:, i])
+            R[i, j] = dps(R[i, j])
+            Q[:, j] -= Q[:, i] * R[i, j]
+
+        Q[:, j] = dps(Q[:, j])
+        if Q[:, j].is_zero_matrix is not True:
+            ranked.append(j)
+            R[j, j] = M.one
+
+    Q = Q.extract(range(Q.rows), ranked)
+    R = R.extract(ranked, range(R.cols))
+
+    if normalize:
+        # Normalization
+        for i in range(Q.cols):
+            norm = Q[:, i].norm()
+            Q[:, i] /= norm
+            R[i, :] *= norm
+
+    return M.__class__(Q), M.__class__(R)
+
+
+def _QRdecomposition(M):
+    r"""Returns a QR decomposition.
+
+    Explanation
+    ===========
+
+    A QR decomposition is a decomposition in the form $A = Q R$
+    where
+
+    - $Q$ is a column orthogonal matrix.
+    - $R$ is a upper triangular (trapezoidal) matrix.
+
+    A column orthogonal matrix satisfies
+    $\mathbb{I} = Q^H Q$ while a full orthogonal matrix satisfies
+    relation $\mathbb{I} = Q Q^H = Q^H Q$ where $I$ is an identity
+    matrix with matching dimensions.
+
+    For matrices which are not square or are rank-deficient, it is
+    sufficient to return a column orthogonal matrix because augmenting
+    them may introduce redundant computations.
+    And an another advantage of this is that you can easily inspect the
+    matrix rank by counting the number of columns of $Q$.
+
+    If you want to augment the results to return a full orthogonal
+    decomposition, you should use the following procedures.
+
+    - Augment the $Q$ matrix with columns that are orthogonal to every
+      other columns and make it square.
+    - Augment the $R$ matrix with zero rows to make it have the same
+      shape as the original matrix.
+
+    The procedure will be illustrated in the examples section.
+
+    Examples
+    ========
+
+    A full rank matrix example:
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([[12, -51, 4], [6, 167, -68], [-4, 24, -41]])
+    >>> Q, R = A.QRdecomposition()
+    >>> Q
+    Matrix([
+    [ 6/7, -69/175, -58/175],
+    [ 3/7, 158/175,   6/175],
+    [-2/7,    6/35,  -33/35]])
+    >>> R
+    Matrix([
+    [14,  21, -14],
+    [ 0, 175, -70],
+    [ 0,   0,  35]])
+
+    If the matrix is square and full rank, the $Q$ matrix becomes
+    orthogonal in both directions, and needs no augmentation.
+
+    >>> Q * Q.H
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+    >>> Q.H * Q
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+
+    >>> A == Q*R
+    True
+
+    A rank deficient matrix example:
+
+    >>> A = Matrix([[12, -51, 0], [6, 167, 0], [-4, 24, 0]])
+    >>> Q, R = A.QRdecomposition()
+    >>> Q
+    Matrix([
+    [ 6/7, -69/175],
+    [ 3/7, 158/175],
+    [-2/7,    6/35]])
+    >>> R
+    Matrix([
+    [14,  21, 0],
+    [ 0, 175, 0]])
+
+    QRdecomposition might return a matrix Q that is rectangular.
+    In this case the orthogonality condition might be satisfied as
+    $\mathbb{I} = Q.H*Q$ but not in the reversed product
+    $\mathbb{I} = Q * Q.H$.
+
+    >>> Q.H * Q
+    Matrix([
+    [1, 0],
+    [0, 1]])
+    >>> Q * Q.H
+    Matrix([
+    [27261/30625,   348/30625, -1914/6125],
+    [  348/30625, 30589/30625,   198/6125],
+    [ -1914/6125,    198/6125,   136/1225]])
+
+    If you want to augment the results to be a full orthogonal
+    decomposition, you should augment $Q$ with an another orthogonal
+    column.
+
+    You are able to append an arbitrary standard basis that are linearly
+    independent to every other columns and you can run the Gram-Schmidt
+    process to make them augmented as orthogonal basis.
+
+    >>> Q_aug = Q.row_join(Matrix([0, 0, 1]))
+    >>> Q_aug = Q_aug.QRdecomposition()[0]
+    >>> Q_aug
+    Matrix([
+    [ 6/7, -69/175, 58/175],
+    [ 3/7, 158/175, -6/175],
+    [-2/7,    6/35,  33/35]])
+    >>> Q_aug.H * Q_aug
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+    >>> Q_aug * Q_aug.H
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+
+    Augmenting the $R$ matrix with zero row is straightforward.
+
+    >>> R_aug = R.col_join(Matrix([[0, 0, 0]]))
+    >>> R_aug
+    Matrix([
+    [14,  21, 0],
+    [ 0, 175, 0],
+    [ 0,   0, 0]])
+    >>> Q_aug * R_aug == A
+    True
+
+    A zero matrix example:
+
+    >>> from sympy import Matrix
+    >>> A = Matrix.zeros(3, 4)
+    >>> Q, R = A.QRdecomposition()
+
+    They may return matrices with zero rows and columns.
+
+    >>> Q
+    Matrix(3, 0, [])
+    >>> R
+    Matrix(0, 4, [])
+    >>> Q*R
+    Matrix([
+    [0, 0, 0, 0],
+    [0, 0, 0, 0],
+    [0, 0, 0, 0]])
+
+    As the same augmentation rule described above, $Q$ can be augmented
+    with columns of an identity matrix and $R$ can be augmented with
+    rows of a zero matrix.
+
+    >>> Q_aug = Q.row_join(Matrix.eye(3))
+    >>> R_aug = R.col_join(Matrix.zeros(3, 4))
+    >>> Q_aug * Q_aug.T
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+    >>> R_aug
+    Matrix([
+    [0, 0, 0, 0],
+    [0, 0, 0, 0],
+    [0, 0, 0, 0]])
+    >>> Q_aug * R_aug == A
+    True
+
+    See Also
+    ========
+
+    sympy.matrices.dense.DenseMatrix.cholesky
+    sympy.matrices.dense.DenseMatrix.LDLdecomposition
+    sympy.matrices.matrices.MatrixBase.LUdecomposition
+    QRsolve
+    """
+    return _QRdecomposition_optional(M, normalize=True)
+
+def _upper_hessenberg_decomposition(A):
+    """Converts a matrix into Hessenberg matrix H.
+
+    Returns 2 matrices H, P s.t.
+    $P H P^{T} = A$, where H is an upper hessenberg matrix
+    and P is an orthogonal matrix
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix
+    >>> A = Matrix([
+    ...     [1,2,3],
+    ...     [-3,5,6],
+    ...     [4,-8,9],
+    ... ])
+    >>> H, P = A.upper_hessenberg_decomposition()
+    >>> H
+    Matrix([
+    [1,    6/5,    17/5],
+    [5, 213/25, -134/25],
+    [0, 216/25,  137/25]])
+    >>> P
+    Matrix([
+    [1,    0,   0],
+    [0, -3/5, 4/5],
+    [0,  4/5, 3/5]])
+    >>> P * H * P.H == A
+    True
+
+
+    References
+    ==========
+
+    .. [#] https://mathworld.wolfram.com/HessenbergDecomposition.html
+    """
+
+    M = A.as_mutable()
+
+    if not M.is_square:
+        raise NonSquareMatrixError("Matrix must be square.")
+
+    n = M.cols
+    P = M.eye(n)
+    H = M
+
+    for j in range(n - 2):
+
+        u = H[j + 1:, j]
+
+        if u[1:, :].is_zero_matrix:
+            continue
+
+        if sign(u[0]) != 0:
+            u[0] = u[0] + sign(u[0]) * u.norm()
+        else:
+            u[0] = u[0] + u.norm()
+
+        v = u / u.norm()
+
+        H[j + 1:, :] = H[j + 1:, :] - 2 * v * (v.H * H[j + 1:, :])
+        H[:, j + 1:] = H[:, j + 1:] - (H[:, j + 1:] * (2 * v)) * v.H
+        P[:, j + 1:] = P[:, j + 1:] - (P[:, j + 1:] * (2 * v)) * v.H
+
+    return H, P

llmeval-env/lib/python3.10/site-packages/sympy/matrices/dense.py

@@ -0,0 +1,1090 @@
+import random
+
+from sympy.core.basic import Basic
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.core.sympify import sympify
+from sympy.functions.elementary.trigonometric import cos, sin
+from sympy.utilities.decorator import doctest_depends_on
+from sympy.utilities.exceptions import sympy_deprecation_warning
+from sympy.utilities.iterables import is_sequence
+
+from .common import ShapeError
+from .decompositions import _cholesky, _LDLdecomposition
+from .matrices import MatrixBase
+from .repmatrix import MutableRepMatrix, RepMatrix
+from .solvers import _lower_triangular_solve, _upper_triangular_solve
+
+
+def _iszero(x):
+    """Returns True if x is zero."""
+    return x.is_zero
+
+
+class DenseMatrix(RepMatrix):
+    """Matrix implementation based on DomainMatrix as the internal representation"""
+
+    #
+    # DenseMatrix is a superclass for both MutableDenseMatrix and
+    # ImmutableDenseMatrix. Methods shared by both classes but not for the
+    # Sparse classes should be implemented here.
+    #
+
+    is_MatrixExpr = False  # type: bool
+
+    _op_priority = 10.01
+    _class_priority = 4
+
+    @property
+    def _mat(self):
+        sympy_deprecation_warning(
+            """
+            The private _mat attribute of Matrix is deprecated. Use the
+            .flat() method instead.
+            """,
+            deprecated_since_version="1.9",
+            active_deprecations_target="deprecated-private-matrix-attributes"
+        )
+
+        return self.flat()
+
+    def _eval_inverse(self, **kwargs):
+        return self.inv(method=kwargs.get('method', 'GE'),
+                        iszerofunc=kwargs.get('iszerofunc', _iszero),
+                        try_block_diag=kwargs.get('try_block_diag', False))
+
+    def as_immutable(self):
+        """Returns an Immutable version of this Matrix
+        """
+        from .immutable import ImmutableDenseMatrix as cls
+        return cls._fromrep(self._rep.copy())
+
+    def as_mutable(self):
+        """Returns a mutable version of this matrix
+
+        Examples
+        ========
+
+        >>> from sympy import ImmutableMatrix
+        >>> X = ImmutableMatrix([[1, 2], [3, 4]])
+        >>> Y = X.as_mutable()
+        >>> Y[1, 1] = 5 # Can set values in Y
+        >>> Y
+        Matrix([
+        [1, 2],
+        [3, 5]])
+        """
+        return Matrix(self)
+
+    def cholesky(self, hermitian=True):
+        return _cholesky(self, hermitian=hermitian)
+
+    def LDLdecomposition(self, hermitian=True):
+        return _LDLdecomposition(self, hermitian=hermitian)
+
+    def lower_triangular_solve(self, rhs):
+        return _lower_triangular_solve(self, rhs)
+
+    def upper_triangular_solve(self, rhs):
+        return _upper_triangular_solve(self, rhs)
+
+    cholesky.__doc__               = _cholesky.__doc__
+    LDLdecomposition.__doc__       = _LDLdecomposition.__doc__
+    lower_triangular_solve.__doc__ = _lower_triangular_solve.__doc__
+    upper_triangular_solve.__doc__ = _upper_triangular_solve.__doc__
+
+
+def _force_mutable(x):
+    """Return a matrix as a Matrix, otherwise return x."""
+    if getattr(x, 'is_Matrix', False):
+        return x.as_mutable()
+    elif isinstance(x, Basic):
+        return x
+    elif hasattr(x, '__array__'):
+        a = x.__array__()
+        if len(a.shape) == 0:
+            return sympify(a)
+        return Matrix(x)
+    return x
+
+
+class MutableDenseMatrix(DenseMatrix, MutableRepMatrix):
+
+    def simplify(self, **kwargs):
+        """Applies simplify to the elements of a matrix in place.
+
+        This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))
+
+        See Also
+        ========
+
+        sympy.simplify.simplify.simplify
+        """
+        from sympy.simplify.simplify import simplify as _simplify
+        for (i, j), element in self.todok().items():
+            self[i, j] = _simplify(element, **kwargs)
+
+
+MutableMatrix = Matrix = MutableDenseMatrix
+
+###########
+# Numpy Utility Functions:
+# list2numpy, matrix2numpy, symmarray
+###########
+
+
+def list2numpy(l, dtype=object):  # pragma: no cover
+    """Converts Python list of SymPy expressions to a NumPy array.
+
+    See Also
+    ========
+
+    matrix2numpy
+    """
+    from numpy import empty
+    a = empty(len(l), dtype)
+    for i, s in enumerate(l):
+        a[i] = s
+    return a
+
+
+def matrix2numpy(m, dtype=object):  # pragma: no cover
+    """Converts SymPy's matrix to a NumPy array.
+
+    See Also
+    ========
+
+    list2numpy
+    """
+    from numpy import empty
+    a = empty(m.shape, dtype)
+    for i in range(m.rows):
+        for j in range(m.cols):
+            a[i, j] = m[i, j]
+    return a
+
+
+###########
+# Rotation matrices:
+# rot_givens, rot_axis[123], rot_ccw_axis[123]
+###########
+
+
+def rot_givens(i, j, theta, dim=3):
+    r"""Returns a a Givens rotation matrix, a a rotation in the
+    plane spanned by two coordinates axes.
+
+    Explanation
+    ===========
+
+    The Givens rotation corresponds to a generalization of rotation
+    matrices to any number of dimensions, given by:
+
+    .. math::
+        G(i, j, \theta) =
+            \begin{bmatrix}
+                1   & \cdots &    0   & \cdots &    0   & \cdots &    0   \\
+                \vdots & \ddots & \vdots &        & \vdots &        & \vdots \\
+                0   & \cdots &    c   & \cdots &   -s   & \cdots &    0   \\
+                \vdots &        & \vdots & \ddots & \vdots &        & \vdots \\
+                0   & \cdots &    s   & \cdots &    c   & \cdots &    0   \\
+                \vdots &        & \vdots &        & \vdots & \ddots & \vdots \\
+                0   & \cdots &    0   & \cdots &    0   & \cdots &    1
+            \end{bmatrix}
+
+    Where $c = \cos(\theta)$ and $s = \sin(\theta)$ appear at the intersections
+    ``i``\th and ``j``\th rows and columns.
+
+    For fixed ``i > j``\, the non-zero elements of a Givens matrix are
+    given by:
+
+    - $g_{kk} = 1$ for $k \ne i,\,j$
+    - $g_{kk} = c$ for $k = i,\,j$
+    - $g_{ji} = -g_{ij} = -s$
+
+    Parameters
+    ==========
+
+    i : int between ``0`` and ``dim - 1``
+        Represents first axis
+    j : int between ``0`` and ``dim - 1``
+        Represents second axis
+    dim : int bigger than 1
+        Number of dimentions. Defaults to 3.
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_givens
+
+    A counterclockwise rotation of pi/3 (60 degrees) around
+    the third axis (z-axis):
+
+    >>> rot_givens(1, 0, pi/3)
+    Matrix([
+    [      1/2, -sqrt(3)/2, 0],
+    [sqrt(3)/2,        1/2, 0],
+    [        0,          0, 1]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_givens(1, 0, pi/2)
+    Matrix([
+    [0, -1, 0],
+    [1,  0, 0],
+    [0,  0, 1]])
+
+    This can be generalized to any number
+    of dimensions:
+
+    >>> rot_givens(1, 0, pi/2, dim=4)
+    Matrix([
+    [0, -1, 0, 0],
+    [1,  0, 0, 0],
+    [0,  0, 1, 0],
+    [0,  0, 0, 1]])
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Givens_rotation
+
+    See Also
+    ========
+
+    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (clockwise around the x axis)
+    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (clockwise around the y axis)
+    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (clockwise around the z axis)
+    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (counterclockwise around the x axis)
+    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (counterclockwise around the y axis)
+    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (counterclockwise around the z axis)
+    """
+    if not isinstance(dim, int) or dim < 2:
+        raise ValueError('dim must be an integer biggen than one, '
+                         'got {}.'.format(dim))
+
+    if i == j:
+        raise ValueError('i and j must be different, '
+                         'got ({}, {})'.format(i, j))
+
+    for ij in [i, j]:
+        if not isinstance(ij, int) or ij < 0 or ij > dim - 1:
+            raise ValueError('i and j must be integers between 0 and '
+                             '{}, got i={} and j={}.'.format(dim-1, i, j))
+
+    theta = sympify(theta)
+    c = cos(theta)
+    s = sin(theta)
+    M = eye(dim)
+    M[i, i] = c
+    M[j, j] = c
+    M[i, j] = s
+    M[j, i] = -s
+    return M
+
+
+def rot_axis3(theta):
+    r"""Returns a rotation matrix for a rotation of theta (in radians)
+    about the 3-axis.
+
+    Explanation
+    ===========
+
+    For a right-handed coordinate system, this corresponds to a
+    clockwise rotation around the `z`-axis, given by:
+
+    .. math::
+
+        R  = \begin{bmatrix}
+                 \cos(\theta) & \sin(\theta) & 0 \\
+                -\sin(\theta) & \cos(\theta) & 0 \\
+                            0 &            0 & 1
+            \end{bmatrix}
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_axis3
+
+    A rotation of pi/3 (60 degrees):
+
+    >>> theta = pi/3
+    >>> rot_axis3(theta)
+    Matrix([
+    [       1/2, sqrt(3)/2, 0],
+    [-sqrt(3)/2,       1/2, 0],
+    [         0,         0, 1]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_axis3(pi/2)
+    Matrix([
+    [ 0, 1, 0],
+    [-1, 0, 0],
+    [ 0, 0, 1]])
+
+    See Also
+    ========
+
+    rot_givens: Returns a Givens rotation matrix (generalized rotation for
+        any number of dimensions)
+    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (counterclockwise around the z axis)
+    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (clockwise around the x axis)
+    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (clockwise around the y axis)
+    """
+    return rot_givens(0, 1, theta, dim=3)
+
+
+def rot_axis2(theta):
+    r"""Returns a rotation matrix for a rotation of theta (in radians)
+    about the 2-axis.
+
+    Explanation
+    ===========
+
+    For a right-handed coordinate system, this corresponds to a
+    clockwise rotation around the `y`-axis, given by:
+
+    .. math::
+
+        R  = \begin{bmatrix}
+                \cos(\theta) & 0 & -\sin(\theta) \\
+                           0 & 1 &             0 \\
+                \sin(\theta) & 0 &  \cos(\theta)
+            \end{bmatrix}
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_axis2
+
+    A rotation of pi/3 (60 degrees):
+
+    >>> theta = pi/3
+    >>> rot_axis2(theta)
+    Matrix([
+    [      1/2, 0, -sqrt(3)/2],
+    [        0, 1,          0],
+    [sqrt(3)/2, 0,        1/2]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_axis2(pi/2)
+    Matrix([
+    [0, 0, -1],
+    [0, 1,  0],
+    [1, 0,  0]])
+
+    See Also
+    ========
+
+    rot_givens: Returns a Givens rotation matrix (generalized rotation for
+        any number of dimensions)
+    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (clockwise around the y axis)
+    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (counterclockwise around the x axis)
+    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (counterclockwise around the z axis)
+    """
+    return rot_givens(2, 0, theta, dim=3)
+
+
+def rot_axis1(theta):
+    r"""Returns a rotation matrix for a rotation of theta (in radians)
+    about the 1-axis.
+
+    Explanation
+    ===========
+
+    For a right-handed coordinate system, this corresponds to a
+    clockwise rotation around the `x`-axis, given by:
+
+    .. math::
+
+        R  = \begin{bmatrix}
+                1 &             0 &            0 \\
+                0 &  \cos(\theta) & \sin(\theta) \\
+                0 & -\sin(\theta) & \cos(\theta)
+            \end{bmatrix}
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_axis1
+
+    A rotation of pi/3 (60 degrees):
+
+    >>> theta = pi/3
+    >>> rot_axis1(theta)
+    Matrix([
+    [1,          0,         0],
+    [0,        1/2, sqrt(3)/2],
+    [0, -sqrt(3)/2,       1/2]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_axis1(pi/2)
+    Matrix([
+    [1,  0, 0],
+    [0,  0, 1],
+    [0, -1, 0]])
+
+    See Also
+    ========
+
+    rot_givens: Returns a Givens rotation matrix (generalized rotation for
+        any number of dimensions)
+    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (counterclockwise around the x axis)
+    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (clockwise around the y axis)
+    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (clockwise around the z axis)
+    """
+    return rot_givens(1, 2, theta, dim=3)
+
+
+def rot_ccw_axis3(theta):
+    r"""Returns a rotation matrix for a rotation of theta (in radians)
+    about the 3-axis.
+
+    Explanation
+    ===========
+
+    For a right-handed coordinate system, this corresponds to a
+    counterclockwise rotation around the `z`-axis, given by:
+
+    .. math::
+
+        R  = \begin{bmatrix}
+                \cos(\theta) & -\sin(\theta) & 0 \\
+                \sin(\theta) &  \cos(\theta) & 0 \\
+                           0 &             0 & 1
+            \end{bmatrix}
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_ccw_axis3
+
+    A rotation of pi/3 (60 degrees):
+
+    >>> theta = pi/3
+    >>> rot_ccw_axis3(theta)
+    Matrix([
+    [      1/2, -sqrt(3)/2, 0],
+    [sqrt(3)/2,        1/2, 0],
+    [        0,          0, 1]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_ccw_axis3(pi/2)
+    Matrix([
+    [0, -1, 0],
+    [1,  0, 0],
+    [0,  0, 1]])
+
+    See Also
+    ========
+
+    rot_givens: Returns a Givens rotation matrix (generalized rotation for
+        any number of dimensions)
+    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (clockwise around the z axis)
+    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (counterclockwise around the x axis)
+    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (counterclockwise around the y axis)
+    """
+    return rot_givens(1, 0, theta, dim=3)
+
+
+def rot_ccw_axis2(theta):
+    r"""Returns a rotation matrix for a rotation of theta (in radians)
+    about the 2-axis.
+
+    Explanation
+    ===========
+
+    For a right-handed coordinate system, this corresponds to a
+    counterclockwise rotation around the `y`-axis, given by:
+
+    .. math::
+
+        R  = \begin{bmatrix}
+                 \cos(\theta) & 0 & \sin(\theta) \\
+                            0 & 1 &            0 \\
+                -\sin(\theta) & 0 & \cos(\theta)
+            \end{bmatrix}
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_ccw_axis2
+
+    A rotation of pi/3 (60 degrees):
+
+    >>> theta = pi/3
+    >>> rot_ccw_axis2(theta)
+    Matrix([
+    [       1/2, 0, sqrt(3)/2],
+    [         0, 1,         0],
+    [-sqrt(3)/2, 0,       1/2]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_ccw_axis2(pi/2)
+    Matrix([
+    [ 0,  0,  1],
+    [ 0,  1,  0],
+    [-1,  0,  0]])
+
+    See Also
+    ========
+
+    rot_givens: Returns a Givens rotation matrix (generalized rotation for
+        any number of dimensions)
+    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (clockwise around the y axis)
+    rot_ccw_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (counterclockwise around the x axis)
+    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (counterclockwise around the z axis)
+    """
+    return rot_givens(0, 2, theta, dim=3)
+
+
+def rot_ccw_axis1(theta):
+    r"""Returns a rotation matrix for a rotation of theta (in radians)
+    about the 1-axis.
+
+    Explanation
+    ===========
+
+    For a right-handed coordinate system, this corresponds to a
+    counterclockwise rotation around the `x`-axis, given by:
+
+    .. math::
+
+        R  = \begin{bmatrix}
+                1 &            0 &             0 \\
+                0 & \cos(\theta) & -\sin(\theta) \\
+                0 & \sin(\theta) &  \cos(\theta)
+            \end{bmatrix}
+
+    Examples
+    ========
+
+    >>> from sympy import pi, rot_ccw_axis1
+
+    A rotation of pi/3 (60 degrees):
+
+    >>> theta = pi/3
+    >>> rot_ccw_axis1(theta)
+    Matrix([
+    [1,         0,          0],
+    [0,       1/2, -sqrt(3)/2],
+    [0, sqrt(3)/2,        1/2]])
+
+    If we rotate by pi/2 (90 degrees):
+
+    >>> rot_ccw_axis1(pi/2)
+    Matrix([
+    [1, 0,  0],
+    [0, 0, -1],
+    [0, 1,  0]])
+
+    See Also
+    ========
+
+    rot_givens: Returns a Givens rotation matrix (generalized rotation for
+        any number of dimensions)
+    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 1-axis (clockwise around the x axis)
+    rot_ccw_axis2: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 2-axis (counterclockwise around the y axis)
+    rot_ccw_axis3: Returns a rotation matrix for a rotation of theta (in radians)
+        about the 3-axis (counterclockwise around the z axis)
+    """
+    return rot_givens(2, 1, theta, dim=3)
+
+
+@doctest_depends_on(modules=('numpy',))
+def symarray(prefix, shape, **kwargs):  # pragma: no cover
+    r"""Create a numpy ndarray of symbols (as an object array).
+
+    The created symbols are named ``prefix_i1_i2_``...  You should thus provide a
+    non-empty prefix if you want your symbols to be unique for different output
+    arrays, as SymPy symbols with identical names are the same object.
+
+    Parameters
+    ----------
+
+    prefix : string
+      A prefix prepended to the name of every symbol.
+
+    shape : int or tuple
+      Shape of the created array.  If an int, the array is one-dimensional; for
+      more than one dimension the shape must be a tuple.
+
+    \*\*kwargs : dict
+      keyword arguments passed on to Symbol
+
+    Examples
+    ========
+    These doctests require numpy.
+
+    >>> from sympy import symarray
+    >>> symarray('', 3)
+    [_0 _1 _2]
+
+    If you want multiple symarrays to contain distinct symbols, you *must*
+    provide unique prefixes:
+
+    >>> a = symarray('', 3)
+    >>> b = symarray('', 3)
+    >>> a[0] == b[0]
+    True
+    >>> a = symarray('a', 3)
+    >>> b = symarray('b', 3)
+    >>> a[0] == b[0]
+    False
+
+    Creating symarrays with a prefix:
+
+    >>> symarray('a', 3)
+    [a_0 a_1 a_2]
+
+    For more than one dimension, the shape must be given as a tuple:
+
+    >>> symarray('a', (2, 3))
+    [[a_0_0 a_0_1 a_0_2]
+     [a_1_0 a_1_1 a_1_2]]
+    >>> symarray('a', (2, 3, 2))
+    [[[a_0_0_0 a_0_0_1]
+      [a_0_1_0 a_0_1_1]
+      [a_0_2_0 a_0_2_1]]
+    <BLANKLINE>
+     [[a_1_0_0 a_1_0_1]
+      [a_1_1_0 a_1_1_1]
+      [a_1_2_0 a_1_2_1]]]
+
+    For setting assumptions of the underlying Symbols:
+
+    >>> [s.is_real for s in symarray('a', 2, real=True)]
+    [True, True]
+    """
+    from numpy import empty, ndindex
+    arr = empty(shape, dtype=object)
+    for index in ndindex(shape):
+        arr[index] = Symbol('%s_%s' % (prefix, '_'.join(map(str, index))),
+                            **kwargs)
+    return arr
+
+
+###############
+# Functions
+###############
+
+def casoratian(seqs, n, zero=True):
+    """Given linear difference operator L of order 'k' and homogeneous
+       equation Ly = 0 we want to compute kernel of L, which is a set
+       of 'k' sequences: a(n), b(n), ... z(n).
+
+       Solutions of L are linearly independent iff their Casoratian,
+       denoted as C(a, b, ..., z), do not vanish for n = 0.
+
+       Casoratian is defined by k x k determinant::
+
+                  +  a(n)     b(n)     . . . z(n)     +
+                  |  a(n+1)   b(n+1)   . . . z(n+1)   |
+                  |    .         .     .        .     |
+                  |    .         .       .      .     |
+                  |    .         .         .    .     |
+                  +  a(n+k-1) b(n+k-1) . . . z(n+k-1) +
+
+       It proves very useful in rsolve_hyper() where it is applied
+       to a generating set of a recurrence to factor out linearly
+       dependent solutions and return a basis:
+
+       >>> from sympy import Symbol, casoratian, factorial
+       >>> n = Symbol('n', integer=True)
+
+       Exponential and factorial are linearly independent:
+
+       >>> casoratian([2**n, factorial(n)], n) != 0
+       True
+
+    """
+
+    seqs = list(map(sympify, seqs))
+
+    if not zero:
+        f = lambda i, j: seqs[j].subs(n, n + i)
+    else:
+        f = lambda i, j: seqs[j].subs(n, i)
+
+    k = len(seqs)
+
+    return Matrix(k, k, f).det()
+
+
+def eye(*args, **kwargs):
+    """Create square identity matrix n x n
+
+    See Also
+    ========
+
+    diag
+    zeros
+    ones
+    """
+
+    return Matrix.eye(*args, **kwargs)
+
+
+def diag(*values, strict=True, unpack=False, **kwargs):
+    """Returns a matrix with the provided values placed on the
+    diagonal. If non-square matrices are included, they will
+    produce a block-diagonal matrix.
+
+    Examples
+    ========
+
+    This version of diag is a thin wrapper to Matrix.diag that differs
+    in that it treats all lists like matrices -- even when a single list
+    is given. If this is not desired, either put a `*` before the list or
+    set `unpack=True`.
+
+    >>> from sympy import diag
+
+    >>> diag([1, 2, 3], unpack=True)  # = diag(1,2,3) or diag(*[1,2,3])
+    Matrix([
+    [1, 0, 0],
+    [0, 2, 0],
+    [0, 0, 3]])
+
+    >>> diag([1, 2, 3])  # a column vector
+    Matrix([
+    [1],
+    [2],
+    [3]])
+
+    See Also
+    ========
+    .common.MatrixCommon.eye
+    .common.MatrixCommon.diagonal
+    .common.MatrixCommon.diag
+    .expressions.blockmatrix.BlockMatrix
+    """
+    return Matrix.diag(*values, strict=strict, unpack=unpack, **kwargs)
+
+
+def GramSchmidt(vlist, orthonormal=False):
+    """Apply the Gram-Schmidt process to a set of vectors.
+
+    Parameters
+    ==========
+
+    vlist : List of Matrix
+        Vectors to be orthogonalized for.
+
+    orthonormal : Bool, optional
+        If true, return an orthonormal basis.
+
+    Returns
+    =======
+
+    vlist : List of Matrix
+        Orthogonalized vectors
+
+    Notes
+    =====
+
+    This routine is mostly duplicate from ``Matrix.orthogonalize``,
+    except for some difference that this always raises error when
+    linearly dependent vectors are found, and the keyword ``normalize``
+    has been named as ``orthonormal`` in this function.
+
+    See Also
+    ========
+
+    .matrices.MatrixSubspaces.orthogonalize
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
+    """
+    return MutableDenseMatrix.orthogonalize(
+        *vlist, normalize=orthonormal, rankcheck=True
+    )
+
+
+def hessian(f, varlist, constraints=()):
+    """Compute Hessian matrix for a function f wrt parameters in varlist
+    which may be given as a sequence or a row/column vector. A list of
+    constraints may optionally be given.
+
+    Examples
+    ========
+
+    >>> from sympy import Function, hessian, pprint
+    >>> from sympy.abc import x, y
+    >>> f = Function('f')(x, y)
+    >>> g1 = Function('g')(x, y)
+    >>> g2 = x**2 + 3*y
+    >>> pprint(hessian(f, (x, y), [g1, g2]))
+    [                   d               d            ]
+    [     0        0    --(g(x, y))     --(g(x, y))  ]
+    [                   dx              dy           ]
+    [                                                ]
+    [     0        0        2*x              3       ]
+    [                                                ]
+    [                     2               2          ]
+    [d                   d               d           ]
+    [--(g(x, y))  2*x   ---(f(x, y))   -----(f(x, y))]
+    [dx                   2            dy dx         ]
+    [                   dx                           ]
+    [                                                ]
+    [                     2               2          ]
+    [d                   d               d           ]
+    [--(g(x, y))   3   -----(f(x, y))   ---(f(x, y)) ]
+    [dy                dy dx              2          ]
+    [                                   dy           ]
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Hessian_matrix
+
+    See Also
+    ========
+
+    sympy.matrices.matrices.MatrixCalculus.jacobian
+    wronskian
+    """
+    # f is the expression representing a function f, return regular matrix
+    if isinstance(varlist, MatrixBase):
+        if 1 not in varlist.shape:
+            raise ShapeError("`varlist` must be a column or row vector.")
+        if varlist.cols == 1:
+            varlist = varlist.T
+        varlist = varlist.tolist()[0]
+    if is_sequence(varlist):
+        n = len(varlist)
+        if not n:
+            raise ShapeError("`len(varlist)` must not be zero.")
+    else:
+        raise ValueError("Improper variable list in hessian function")
+    if not getattr(f, 'diff'):
+        # check differentiability
+        raise ValueError("Function `f` (%s) is not differentiable" % f)
+    m = len(constraints)
+    N = m + n
+    out = zeros(N)
+    for k, g in enumerate(constraints):
+        if not getattr(g, 'diff'):
+            # check differentiability
+            raise ValueError("Function `f` (%s) is not differentiable" % f)
+        for i in range(n):
+            out[k, i + m] = g.diff(varlist[i])
+    for i in range(n):
+        for j in range(i, n):
+            out[i + m, j + m] = f.diff(varlist[i]).diff(varlist[j])
+    for i in range(N):
+        for j in range(i + 1, N):
+            out[j, i] = out[i, j]
+    return out
+
+
+def jordan_cell(eigenval, n):
+    """
+    Create a Jordan block:
+
+    Examples
+    ========
+
+    >>> from sympy import jordan_cell
+    >>> from sympy.abc import x
+    >>> jordan_cell(x, 4)
+    Matrix([
+    [x, 1, 0, 0],
+    [0, x, 1, 0],
+    [0, 0, x, 1],
+    [0, 0, 0, x]])
+    """
+
+    return Matrix.jordan_block(size=n, eigenvalue=eigenval)
+
+
+def matrix_multiply_elementwise(A, B):
+    """Return the Hadamard product (elementwise product) of A and B
+
+    >>> from sympy import Matrix, matrix_multiply_elementwise
+    >>> A = Matrix([[0, 1, 2], [3, 4, 5]])
+    >>> B = Matrix([[1, 10, 100], [100, 10, 1]])
+    >>> matrix_multiply_elementwise(A, B)
+    Matrix([
+    [  0, 10, 200],
+    [300, 40,   5]])
+
+    See Also
+    ========
+
+    sympy.matrices.common.MatrixCommon.__mul__
+    """
+    return A.multiply_elementwise(B)
+
+
+def ones(*args, **kwargs):
+    """Returns a matrix of ones with ``rows`` rows and ``cols`` columns;
+    if ``cols`` is omitted a square matrix will be returned.
+
+    See Also
+    ========
+
+    zeros
+    eye
+    diag
+    """
+
+    if 'c' in kwargs:
+        kwargs['cols'] = kwargs.pop('c')
+
+    return Matrix.ones(*args, **kwargs)
+
+
+def randMatrix(r, c=None, min=0, max=99, seed=None, symmetric=False,
+               percent=100, prng=None):
+    """Create random matrix with dimensions ``r`` x ``c``. If ``c`` is omitted
+    the matrix will be square. If ``symmetric`` is True the matrix must be
+    square. If ``percent`` is less than 100 then only approximately the given
+    percentage of elements will be non-zero.
+
+    The pseudo-random number generator used to generate matrix is chosen in the
+    following way.
+
+    * If ``prng`` is supplied, it will be used as random number generator.
+      It should be an instance of ``random.Random``, or at least have
+      ``randint`` and ``shuffle`` methods with same signatures.
+    * if ``prng`` is not supplied but ``seed`` is supplied, then new
+      ``random.Random`` with given ``seed`` will be created;
+    * otherwise, a new ``random.Random`` with default seed will be used.
+
+    Examples
+    ========
+
+    >>> from sympy import randMatrix
+    >>> randMatrix(3) # doctest:+SKIP
+    [25, 45, 27]
+    [44, 54,  9]
+    [23, 96, 46]
+    >>> randMatrix(3, 2) # doctest:+SKIP
+    [87, 29]
+    [23, 37]
+    [90, 26]
+    >>> randMatrix(3, 3, 0, 2) # doctest:+SKIP
+    [0, 2, 0]
+    [2, 0, 1]
+    [0, 0, 1]
+    >>> randMatrix(3, symmetric=True) # doctest:+SKIP
+    [85, 26, 29]
+    [26, 71, 43]
+    [29, 43, 57]
+    >>> A = randMatrix(3, seed=1)
+    >>> B = randMatrix(3, seed=2)
+    >>> A == B
+    False
+    >>> A == randMatrix(3, seed=1)
+    True
+    >>> randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP
+    [77, 70,  0],
+    [70,  0,  0],
+    [ 0,  0, 88]
+    """
+    # Note that ``Random()`` is equivalent to ``Random(None)``
+    prng = prng or random.Random(seed)
+
+    if c is None:
+        c = r
+
+    if symmetric and r != c:
+        raise ValueError('For symmetric matrices, r must equal c, but %i != %i' % (r, c))
+
+    ij = range(r * c)
+    if percent != 100:
+        ij = prng.sample(ij, int(len(ij)*percent // 100))
+
+    m = zeros(r, c)
+
+    if not symmetric:
+        for ijk in ij:
+            i, j = divmod(ijk, c)
+            m[i, j] = prng.randint(min, max)
+    else:
+        for ijk in ij:
+            i, j = divmod(ijk, c)
+            if i <= j:
+                m[i, j] = m[j, i] = prng.randint(min, max)
+
+    return m
+
+
+def wronskian(functions, var, method='bareiss'):
+    """
+    Compute Wronskian for [] of functions
+
+    ::
+
+                         | f1       f2        ...   fn      |
+                         | f1'      f2'       ...   fn'     |
+                         |  .        .        .      .      |
+        W(f1, ..., fn) = |  .        .         .     .      |
+                         |  .        .          .    .      |
+                         |  (n)      (n)            (n)     |
+                         | D   (f1) D   (f2)  ...  D   (fn) |
+
+    see: https://en.wikipedia.org/wiki/Wronskian
+
+    See Also
+    ========
+
+    sympy.matrices.matrices.MatrixCalculus.jacobian
+    hessian
+    """
+
+    functions = [sympify(f) for f in functions]
+    n = len(functions)
+    if n == 0:
+        return S.One
+    W = Matrix(n, n, lambda i, j: functions[i].diff(var, j))
+    return W.det(method)
+
+
+def zeros(*args, **kwargs):
+    """Returns a matrix of zeros with ``rows`` rows and ``cols`` columns;
+    if ``cols`` is omitted a square matrix will be returned.
+
+    See Also
+    ========
+
+    ones
+    eye
+    diag
+    """
+
+    if 'c' in kwargs:
+        kwargs['cols'] = kwargs.pop('c')
+
+    return Matrix.zeros(*args, **kwargs)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/__init__.py

@@ -0,0 +1,62 @@
+""" A module which handles Matrix Expressions """
+
+from .slice import MatrixSlice
+from .blockmatrix import BlockMatrix, BlockDiagMatrix, block_collapse, blockcut
+from .companion import CompanionMatrix
+from .funcmatrix import FunctionMatrix
+from .inverse import Inverse
+from .matadd import MatAdd
+from .matexpr import MatrixExpr, MatrixSymbol, matrix_symbols
+from .matmul import MatMul
+from .matpow import MatPow
+from .trace import Trace, trace
+from .determinant import Determinant, det, Permanent, per
+from .transpose import Transpose
+from .adjoint import Adjoint
+from .hadamard import hadamard_product, HadamardProduct, hadamard_power, HadamardPower
+from .diagonal import DiagonalMatrix, DiagonalOf, DiagMatrix, diagonalize_vector
+from .dotproduct import DotProduct
+from .kronecker import kronecker_product, KroneckerProduct, combine_kronecker
+from .permutation import PermutationMatrix, MatrixPermute
+from .sets import MatrixSet
+from .special import ZeroMatrix, Identity, OneMatrix
+
+__all__ = [
+    'MatrixSlice',
+
+    'BlockMatrix', 'BlockDiagMatrix', 'block_collapse', 'blockcut',
+    'FunctionMatrix',
+
+    'CompanionMatrix',
+
+    'Inverse',
+
+    'MatAdd',
+
+    'Identity', 'MatrixExpr', 'MatrixSymbol', 'ZeroMatrix', 'OneMatrix',
+    'matrix_symbols', 'MatrixSet',
+
+    'MatMul',
+
+    'MatPow',
+
+    'Trace', 'trace',
+
+    'Determinant', 'det',
+
+    'Transpose',
+
+    'Adjoint',
+
+    'hadamard_product', 'HadamardProduct', 'hadamard_power', 'HadamardPower',
+
+    'DiagonalMatrix', 'DiagonalOf', 'DiagMatrix', 'diagonalize_vector',
+
+    'DotProduct',
+
+    'kronecker_product', 'KroneckerProduct', 'combine_kronecker',
+
+    'PermutationMatrix', 'MatrixPermute',
+
+    'Permanent', 'per'
+]

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/_shape.py

@@ -0,0 +1,102 @@
+from sympy.core.relational import Eq
+from sympy.core.expr import Expr
+from sympy.core.numbers import Integer
+from sympy.logic.boolalg import Boolean, And
+from sympy.matrices.expressions.matexpr import MatrixExpr
+from sympy.matrices.common import ShapeError
+from typing import Union
+
+
+def is_matadd_valid(*args: MatrixExpr) -> Boolean:
+    """Return the symbolic condition how ``MatAdd``, ``HadamardProduct``
+    makes sense.
+
+    Parameters
+    ==========
+
+    args
+        The list of arguments of matrices to be tested for.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, symbols
+    >>> from sympy.matrices.expressions._shape import is_matadd_valid
+
+    >>> m, n, p, q = symbols('m n p q')
+    >>> A = MatrixSymbol('A', m, n)
+    >>> B = MatrixSymbol('B', p, q)
+    >>> is_matadd_valid(A, B)
+    Eq(m, p) & Eq(n, q)
+    """
+    rows, cols = zip(*(arg.shape for arg in args))
+    return And(
+        *(Eq(i, j) for i, j in zip(rows[:-1], rows[1:])),
+        *(Eq(i, j) for i, j in zip(cols[:-1], cols[1:])),
+    )
+
+
+def is_matmul_valid(*args: Union[MatrixExpr, Expr]) -> Boolean:
+    """Return the symbolic condition how ``MatMul`` makes sense
+
+    Parameters
+    ==========
+
+    args
+        The list of arguments of matrices and scalar expressions to be tested
+        for.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, symbols
+    >>> from sympy.matrices.expressions._shape import is_matmul_valid
+
+    >>> m, n, p, q = symbols('m n p q')
+    >>> A = MatrixSymbol('A', m, n)
+    >>> B = MatrixSymbol('B', p, q)
+    >>> is_matmul_valid(A, B)
+    Eq(n, p)
+    """
+    rows, cols = zip(*(arg.shape for arg in args if isinstance(arg, MatrixExpr)))
+    return And(*(Eq(i, j) for i, j in zip(cols[:-1], rows[1:])))
+
+
+def is_square(arg: MatrixExpr, /) -> Boolean:
+    """Return the symbolic condition how the matrix is assumed to be square
+
+    Parameters
+    ==========
+
+    arg
+        The matrix to be tested for.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, symbols
+    >>> from sympy.matrices.expressions._shape import is_square
+
+    >>> m, n = symbols('m n')
+    >>> A = MatrixSymbol('A', m, n)
+    >>> is_square(A)
+    Eq(m, n)
+    """
+    return Eq(arg.rows, arg.cols)
+
+
+def validate_matadd_integer(*args: MatrixExpr) -> None:
+    """Validate matrix shape for addition only for integer values"""
+    rows, cols = zip(*(x.shape for x in args))
+    if len(set(filter(lambda x: isinstance(x, (int, Integer)), rows))) > 1:
+        raise ShapeError(f"Matrices have mismatching shape: {rows}")
+    if len(set(filter(lambda x: isinstance(x, (int, Integer)), cols))) > 1:
+        raise ShapeError(f"Matrices have mismatching shape: {cols}")
+
+
+def validate_matmul_integer(*args: MatrixExpr) -> None:
+    """Validate matrix shape for multiplication only for integer values"""
+    for A, B in zip(args[:-1], args[1:]):
+        i, j = A.cols, B.rows
+        if isinstance(i, (int, Integer)) and isinstance(j, (int, Integer)) and i != j:
+            raise ShapeError("Matrices are not aligned", i, j)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/adjoint.py

@@ -0,0 +1,61 @@
+from sympy.core import Basic
+from sympy.functions import adjoint, conjugate
+from sympy.matrices.expressions.transpose import transpose
+from sympy.matrices.expressions.matexpr import MatrixExpr
+
+
+class Adjoint(MatrixExpr):
+    """
+    The Hermitian adjoint of a matrix expression.
+
+    This is a symbolic object that simply stores its argument without
+    evaluating it. To actually compute the adjoint, use the ``adjoint()``
+    function.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, Adjoint, adjoint
+    >>> A = MatrixSymbol('A', 3, 5)
+    >>> B = MatrixSymbol('B', 5, 3)
+    >>> Adjoint(A*B)
+    Adjoint(A*B)
+    >>> adjoint(A*B)
+    Adjoint(B)*Adjoint(A)
+    >>> adjoint(A*B) == Adjoint(A*B)
+    False
+    >>> adjoint(A*B) == Adjoint(A*B).doit()
+    True
+    """
+    is_Adjoint = True
+
+    def doit(self, **hints):
+        arg = self.arg
+        if hints.get('deep', True) and isinstance(arg, Basic):
+            return adjoint(arg.doit(**hints))
+        else:
+            return adjoint(self.arg)
+
+    @property
+    def arg(self):
+        return self.args[0]
+
+    @property
+    def shape(self):
+        return self.arg.shape[::-1]
+
+    def _entry(self, i, j, **kwargs):
+        return conjugate(self.arg._entry(j, i, **kwargs))
+
+    def _eval_adjoint(self):
+        return self.arg
+
+    def _eval_conjugate(self):
+        return transpose(self.arg)
+
+    def _eval_trace(self):
+        from sympy.matrices.expressions.trace import Trace
+        return conjugate(Trace(self.arg))
+
+    def _eval_transpose(self):
+        return conjugate(self.arg)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/applyfunc.py

@@ -0,0 +1,204 @@
+from sympy.core.expr import ExprBuilder
+from sympy.core.function import (Function, FunctionClass, Lambda)
+from sympy.core.symbol import Dummy
+from sympy.core.sympify import sympify, _sympify
+from sympy.matrices.expressions import MatrixExpr
+from sympy.matrices.matrices import MatrixBase
+
+
+class ElementwiseApplyFunction(MatrixExpr):
+    r"""
+    Apply function to a matrix elementwise without evaluating.
+
+    Examples
+    ========
+
+    It can be created by calling ``.applyfunc(<function>)`` on a matrix
+    expression:
+
+    >>> from sympy import MatrixSymbol
+    >>> from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
+    >>> from sympy import exp
+    >>> X = MatrixSymbol("X", 3, 3)
+    >>> X.applyfunc(exp)
+    Lambda(_d, exp(_d)).(X)
+
+    Otherwise using the class constructor:
+
+    >>> from sympy import eye
+    >>> expr = ElementwiseApplyFunction(exp, eye(3))
+    >>> expr
+    Lambda(_d, exp(_d)).(Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]]))
+    >>> expr.doit()
+    Matrix([
+    [E, 1, 1],
+    [1, E, 1],
+    [1, 1, E]])
+
+    Notice the difference with the real mathematical functions:
+
+    >>> exp(eye(3))
+    Matrix([
+    [E, 0, 0],
+    [0, E, 0],
+    [0, 0, E]])
+    """
+
+    def __new__(cls, function, expr):
+        expr = _sympify(expr)
+        if not expr.is_Matrix:
+            raise ValueError("{} must be a matrix instance.".format(expr))
+
+        if expr.shape == (1, 1):
+            # Check if the function returns a matrix, in that case, just apply
+            # the function instead of creating an ElementwiseApplyFunc object:
+            ret = function(expr)
+            if isinstance(ret, MatrixExpr):
+                return ret
+
+        if not isinstance(function, (FunctionClass, Lambda)):
+            d = Dummy('d')
+            function = Lambda(d, function(d))
+
+        function = sympify(function)
+        if not isinstance(function, (FunctionClass, Lambda)):
+            raise ValueError(
+                "{} should be compatible with SymPy function classes."
+                .format(function))
+
+        if 1 not in function.nargs:
+            raise ValueError(
+                '{} should be able to accept 1 arguments.'.format(function))
+
+        if not isinstance(function, Lambda):
+            d = Dummy('d')
+            function = Lambda(d, function(d))
+
+        obj = MatrixExpr.__new__(cls, function, expr)
+        return obj
+
+    @property
+    def function(self):
+        return self.args[0]
+
+    @property
+    def expr(self):
+        return self.args[1]
+
+    @property
+    def shape(self):
+        return self.expr.shape
+
+    def doit(self, **hints):
+        deep = hints.get("deep", True)
+        expr = self.expr
+        if deep:
+            expr = expr.doit(**hints)
+        function = self.function
+        if isinstance(function, Lambda) and function.is_identity:
+            # This is a Lambda containing the identity function.
+            return expr
+        if isinstance(expr, MatrixBase):
+            return expr.applyfunc(self.function)
+        elif isinstance(expr, ElementwiseApplyFunction):
+            return ElementwiseApplyFunction(
+                lambda x: self.function(expr.function(x)),
+                expr.expr
+            ).doit(**hints)
+        else:
+            return self
+
+    def _entry(self, i, j, **kwargs):
+        return self.function(self.expr._entry(i, j, **kwargs))
+
+    def _get_function_fdiff(self):
+        d = Dummy("d")
+        function = self.function(d)
+        fdiff = function.diff(d)
+        if isinstance(fdiff, Function):
+            fdiff = type(fdiff)
+        else:
+            fdiff = Lambda(d, fdiff)
+        return fdiff
+
+    def _eval_derivative(self, x):
+        from sympy.matrices.expressions.hadamard import hadamard_product
+        dexpr = self.expr.diff(x)
+        fdiff = self._get_function_fdiff()
+        return hadamard_product(
+            dexpr,
+            ElementwiseApplyFunction(fdiff, self.expr)
+        )
+
+    def _eval_derivative_matrix_lines(self, x):
+        from sympy.matrices.expressions.special import Identity
+        from sympy.tensor.array.expressions.array_expressions import ArrayContraction
+        from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
+        from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
+
+        fdiff = self._get_function_fdiff()
+        lr = self.expr._eval_derivative_matrix_lines(x)
+        ewdiff = ElementwiseApplyFunction(fdiff, self.expr)
+        if 1 in x.shape:
+            # Vector:
+            iscolumn = self.shape[1] == 1
+            for i in lr:
+                if iscolumn:
+                    ptr1 = i.first_pointer
+                    ptr2 = Identity(self.shape[1])
+                else:
+                    ptr1 = Identity(self.shape[0])
+                    ptr2 = i.second_pointer
+
+                subexpr = ExprBuilder(
+                    ArrayDiagonal,
+                    [
+                        ExprBuilder(
+                            ArrayTensorProduct,
+                            [
+                                ewdiff,
+                                ptr1,
+                                ptr2,
+                            ]
+                        ),
+                        (0, 2) if iscolumn else (1, 4)
+                    ],
+                    validator=ArrayDiagonal._validate
+                )
+                i._lines = [subexpr]
+                i._first_pointer_parent = subexpr.args[0].args
+                i._first_pointer_index = 1
+                i._second_pointer_parent = subexpr.args[0].args
+                i._second_pointer_index = 2
+        else:
+            # Matrix case:
+            for i in lr:
+                ptr1 = i.first_pointer
+                ptr2 = i.second_pointer
+                newptr1 = Identity(ptr1.shape[1])
+                newptr2 = Identity(ptr2.shape[1])
+                subexpr = ExprBuilder(
+                    ArrayContraction,
+                    [
+                        ExprBuilder(
+                            ArrayTensorProduct,
+                            [ptr1, newptr1, ewdiff, ptr2, newptr2]
+                        ),
+                        (1, 2, 4),
+                        (5, 7, 8),
+                    ],
+                    validator=ArrayContraction._validate
+                )
+                i._first_pointer_parent = subexpr.args[0].args
+                i._first_pointer_index = 1
+                i._second_pointer_parent = subexpr.args[0].args
+                i._second_pointer_index = 4
+                i._lines = [subexpr]
+        return lr
+
+    def _eval_transpose(self):
+        from sympy.matrices.expressions.transpose import Transpose
+        return self.func(self.function, Transpose(self.expr).doit())

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/blockmatrix.py

@@ -0,0 +1,979 @@
+from sympy.assumptions.ask import (Q, ask)
+from sympy.core import Basic, Add, Mul, S
+from sympy.core.sympify import _sympify
+from sympy.functions import adjoint
+from sympy.functions.elementary.complexes import re, im
+from sympy.strategies import typed, exhaust, condition, do_one, unpack
+from sympy.strategies.traverse import bottom_up
+from sympy.utilities.iterables import is_sequence, sift
+from sympy.utilities.misc import filldedent
+
+from sympy.matrices import Matrix, ShapeError
+from sympy.matrices.common import NonInvertibleMatrixError
+from sympy.matrices.expressions.determinant import det, Determinant
+from sympy.matrices.expressions.inverse import Inverse
+from sympy.matrices.expressions.matadd import MatAdd
+from sympy.matrices.expressions.matexpr import MatrixExpr, MatrixElement
+from sympy.matrices.expressions.matmul import MatMul
+from sympy.matrices.expressions.matpow import MatPow
+from sympy.matrices.expressions.slice import MatrixSlice
+from sympy.matrices.expressions.special import ZeroMatrix, Identity
+from sympy.matrices.expressions.trace import trace
+from sympy.matrices.expressions.transpose import Transpose, transpose
+
+
+class BlockMatrix(MatrixExpr):
+    """A BlockMatrix is a Matrix comprised of other matrices.
+
+    The submatrices are stored in a SymPy Matrix object but accessed as part of
+    a Matrix Expression
+
+    >>> from sympy import (MatrixSymbol, BlockMatrix, symbols,
+    ...     Identity, ZeroMatrix, block_collapse)
+    >>> n,m,l = symbols('n m l')
+    >>> X = MatrixSymbol('X', n, n)
+    >>> Y = MatrixSymbol('Y', m, m)
+    >>> Z = MatrixSymbol('Z', n, m)
+    >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
+    >>> print(B)
+    Matrix([
+    [X, Z],
+    [0, Y]])
+
+    >>> C = BlockMatrix([[Identity(n), Z]])
+    >>> print(C)
+    Matrix([[I, Z]])
+
+    >>> print(block_collapse(C*B))
+    Matrix([[X, Z + Z*Y]])
+
+    Some matrices might be comprised of rows of blocks with
+    the matrices in each row having the same height and the
+    rows all having the same total number of columns but
+    not having the same number of columns for each matrix
+    in each row. In this case, the matrix is not a block
+    matrix and should be instantiated by Matrix.
+
+    >>> from sympy import ones, Matrix
+    >>> dat = [
+    ... [ones(3,2), ones(3,3)*2],
+    ... [ones(2,3)*3, ones(2,2)*4]]
+    ...
+    >>> BlockMatrix(dat)
+    Traceback (most recent call last):
+    ...
+    ValueError:
+    Although this matrix is comprised of blocks, the blocks do not fill
+    the matrix in a size-symmetric fashion. To create a full matrix from
+    these arguments, pass them directly to Matrix.
+    >>> Matrix(dat)
+    Matrix([
+    [1, 1, 2, 2, 2],
+    [1, 1, 2, 2, 2],
+    [1, 1, 2, 2, 2],
+    [3, 3, 3, 4, 4],
+    [3, 3, 3, 4, 4]])
+
+    See Also
+    ========
+    sympy.matrices.matrices.MatrixBase.irregular
+    """
+    def __new__(cls, *args, **kwargs):
+        from sympy.matrices.immutable import ImmutableDenseMatrix
+        isMat = lambda i: getattr(i, 'is_Matrix', False)
+        if len(args) != 1 or \
+                not is_sequence(args[0]) or \
+                len({isMat(r) for r in args[0]}) != 1:
+            raise ValueError(filldedent('''
+                expecting a sequence of 1 or more rows
+                containing Matrices.'''))
+        rows = args[0] if args else []
+        if not isMat(rows):
+            if rows and isMat(rows[0]):
+                rows = [rows]  # rows is not list of lists or []
+            # regularity check
+            # same number of matrices in each row
+            blocky = ok = len({len(r) for r in rows}) == 1
+            if ok:
+                # same number of rows for each matrix in a row
+                for r in rows:
+                    ok = len({i.rows for i in r}) == 1
+                    if not ok:
+                        break
+                blocky = ok
+                if ok:
+                    # same number of cols for each matrix in each col
+                    for c in range(len(rows[0])):
+                        ok = len({rows[i][c].cols
+                            for i in range(len(rows))}) == 1
+                        if not ok:
+                            break
+            if not ok:
+                # same total cols in each row
+                ok = len({
+                    sum([i.cols for i in r]) for r in rows}) == 1
+                if blocky and ok:
+                    raise ValueError(filldedent('''
+                        Although this matrix is comprised of blocks,
+                        the blocks do not fill the matrix in a
+                        size-symmetric fashion. To create a full matrix
+                        from these arguments, pass them directly to
+                        Matrix.'''))
+                raise ValueError(filldedent('''
+                    When there are not the same number of rows in each
+                    row's matrices or there are not the same number of
+                    total columns in each row, the matrix is not a
+                    block matrix. If this matrix is known to consist of
+                    blocks fully filling a 2-D space then see
+                    Matrix.irregular.'''))
+        mat = ImmutableDenseMatrix(rows, evaluate=False)
+        obj = Basic.__new__(cls, mat)
+        return obj
+
+    @property
+    def shape(self):
+        numrows = numcols = 0
+        M = self.blocks
+        for i in range(M.shape[0]):
+            numrows += M[i, 0].shape[0]
+        for i in range(M.shape[1]):
+            numcols += M[0, i].shape[1]
+        return (numrows, numcols)
+
+    @property
+    def blockshape(self):
+        return self.blocks.shape
+
+    @property
+    def blocks(self):
+        return self.args[0]
+
+    @property
+    def rowblocksizes(self):
+        return [self.blocks[i, 0].rows for i in range(self.blockshape[0])]
+
+    @property
+    def colblocksizes(self):
+        return [self.blocks[0, i].cols for i in range(self.blockshape[1])]
+
+    def structurally_equal(self, other):
+        return (isinstance(other, BlockMatrix)
+            and self.shape == other.shape
+            and self.blockshape == other.blockshape
+            and self.rowblocksizes == other.rowblocksizes
+            and self.colblocksizes == other.colblocksizes)
+
+    def _blockmul(self, other):
+        if (isinstance(other, BlockMatrix) and
+                self.colblocksizes == other.rowblocksizes):
+            return BlockMatrix(self.blocks*other.blocks)
+
+        return self * other
+
+    def _blockadd(self, other):
+        if (isinstance(other, BlockMatrix)
+                and self.structurally_equal(other)):
+            return BlockMatrix(self.blocks + other.blocks)
+
+        return self + other
+
+    def _eval_transpose(self):
+        # Flip all the individual matrices
+        matrices = [transpose(matrix) for matrix in self.blocks]
+        # Make a copy
+        M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
+        # Transpose the block structure
+        M = M.transpose()
+        return BlockMatrix(M)
+
+    def _eval_adjoint(self):
+        # Adjoint all the individual matrices
+        matrices = [adjoint(matrix) for matrix in self.blocks]
+        # Make a copy
+        M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
+        # Transpose the block structure
+        M = M.transpose()
+        return BlockMatrix(M)
+
+    def _eval_trace(self):
+        if self.rowblocksizes == self.colblocksizes:
+            return Add(*[trace(self.blocks[i, i])
+                        for i in range(self.blockshape[0])])
+        raise NotImplementedError(
+            "Can't perform trace of irregular blockshape")
+
+    def _eval_determinant(self):
+        if self.blockshape == (1, 1):
+            return det(self.blocks[0, 0])
+        if self.blockshape == (2, 2):
+            [[A, B],
+             [C, D]] = self.blocks.tolist()
+            if ask(Q.invertible(A)):
+                return det(A)*det(D - C*A.I*B)
+            elif ask(Q.invertible(D)):
+                return det(D)*det(A - B*D.I*C)
+        return Determinant(self)
+
+    def _eval_as_real_imag(self):
+        real_matrices = [re(matrix) for matrix in self.blocks]
+        real_matrices = Matrix(self.blockshape[0], self.blockshape[1], real_matrices)
+
+        im_matrices = [im(matrix) for matrix in self.blocks]
+        im_matrices = Matrix(self.blockshape[0], self.blockshape[1], im_matrices)
+
+        return (BlockMatrix(real_matrices), BlockMatrix(im_matrices))
+
+    def transpose(self):
+        """Return transpose of matrix.
+
+        Examples
+        ========
+
+        >>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix
+        >>> from sympy.abc import m, n
+        >>> X = MatrixSymbol('X', n, n)
+        >>> Y = MatrixSymbol('Y', m, m)
+        >>> Z = MatrixSymbol('Z', n, m)
+        >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
+        >>> B.transpose()
+        Matrix([
+        [X.T,  0],
+        [Z.T, Y.T]])
+        >>> _.transpose()
+        Matrix([
+        [X, Z],
+        [0, Y]])
+        """
+        return self._eval_transpose()
+
+    def schur(self, mat = 'A', generalized = False):
+        """Return the Schur Complement of the 2x2 BlockMatrix
+
+        Parameters
+        ==========
+
+        mat : String, optional
+            The matrix with respect to which the
+            Schur Complement is calculated. 'A' is
+            used by default
+
+        generalized : bool, optional
+            If True, returns the generalized Schur
+            Component which uses Moore-Penrose Inverse
+
+        Examples
+        ========
+
+        >>> from sympy import symbols, MatrixSymbol, BlockMatrix
+        >>> m, n = symbols('m n')
+        >>> A = MatrixSymbol('A', n, n)
+        >>> B = MatrixSymbol('B', n, m)
+        >>> C = MatrixSymbol('C', m, n)
+        >>> D = MatrixSymbol('D', m, m)
+        >>> X = BlockMatrix([[A, B], [C, D]])
+
+        The default Schur Complement is evaluated with "A"
+
+        >>> X.schur()
+        -C*A**(-1)*B + D
+        >>> X.schur('D')
+        A - B*D**(-1)*C
+
+        Schur complement with non-invertible matrices is not
+        defined. Instead, the generalized Schur complement can
+        be calculated which uses the Moore-Penrose Inverse. To
+        achieve this, `generalized` must be set to `True`
+
+        >>> X.schur('B', generalized=True)
+        C - D*(B.T*B)**(-1)*B.T*A
+        >>> X.schur('C', generalized=True)
+        -A*(C.T*C)**(-1)*C.T*D + B
+
+        Returns
+        =======
+
+        M : Matrix
+            The Schur Complement Matrix
+
+        Raises
+        ======
+
+        ShapeError
+            If the block matrix is not a 2x2 matrix
+
+        NonInvertibleMatrixError
+            If given matrix is non-invertible
+
+        References
+        ==========
+
+        .. [1] Wikipedia Article on Schur Component : https://en.wikipedia.org/wiki/Schur_complement
+
+        See Also
+        ========
+
+        sympy.matrices.matrices.MatrixBase.pinv
+        """
+
+        if self.blockshape == (2, 2):
+            [[A, B],
+             [C, D]] = self.blocks.tolist()
+            d={'A' : A, 'B' : B, 'C' : C, 'D' : D}
+            try:
+                inv = (d[mat].T*d[mat]).inv()*d[mat].T if generalized else d[mat].inv()
+                if mat == 'A':
+                    return D - C * inv * B
+                elif mat == 'B':
+                    return C - D * inv * A
+                elif mat == 'C':
+                    return B - A * inv * D
+                elif mat == 'D':
+                    return A - B * inv * C
+                #For matrices where no sub-matrix is square
+                return self
+            except NonInvertibleMatrixError:
+                raise NonInvertibleMatrixError('The given matrix is not invertible. Please set generalized=True \
+            to compute the generalized Schur Complement which uses Moore-Penrose Inverse')
+        else:
+            raise ShapeError('Schur Complement can only be calculated for 2x2 block matrices')
+
+    def LDUdecomposition(self):
+        """Returns the Block LDU decomposition of
+        a 2x2 Block Matrix
+
+        Returns
+        =======
+
+        (L, D, U) : Matrices
+            L : Lower Diagonal Matrix
+            D : Diagonal Matrix
+            U : Upper Diagonal Matrix
+
+        Examples
+        ========
+
+        >>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
+        >>> m, n = symbols('m n')
+        >>> A = MatrixSymbol('A', n, n)
+        >>> B = MatrixSymbol('B', n, m)
+        >>> C = MatrixSymbol('C', m, n)
+        >>> D = MatrixSymbol('D', m, m)
+        >>> X = BlockMatrix([[A, B], [C, D]])
+        >>> L, D, U = X.LDUdecomposition()
+        >>> block_collapse(L*D*U)
+        Matrix([
+        [A, B],
+        [C, D]])
+
+        Raises
+        ======
+
+        ShapeError
+            If the block matrix is not a 2x2 matrix
+
+        NonInvertibleMatrixError
+            If the matrix "A" is non-invertible
+
+        See Also
+        ========
+        sympy.matrices.expressions.blockmatrix.BlockMatrix.UDLdecomposition
+        sympy.matrices.expressions.blockmatrix.BlockMatrix.LUdecomposition
+        """
+        if self.blockshape == (2,2):
+            [[A, B],
+             [C, D]] = self.blocks.tolist()
+            try:
+                AI = A.I
+            except NonInvertibleMatrixError:
+                raise NonInvertibleMatrixError('Block LDU decomposition cannot be calculated when\
+                    "A" is singular')
+            Ip = Identity(B.shape[0])
+            Iq = Identity(B.shape[1])
+            Z = ZeroMatrix(*B.shape)
+            L = BlockMatrix([[Ip, Z], [C*AI, Iq]])
+            D = BlockDiagMatrix(A, self.schur())
+            U = BlockMatrix([[Ip, AI*B],[Z.T, Iq]])
+            return L, D, U
+        else:
+            raise ShapeError("Block LDU decomposition is supported only for 2x2 block matrices")
+
+    def UDLdecomposition(self):
+        """Returns the Block UDL decomposition of
+        a 2x2 Block Matrix
+
+        Returns
+        =======
+
+        (U, D, L) : Matrices
+            U : Upper Diagonal Matrix
+            D : Diagonal Matrix
+            L : Lower Diagonal Matrix
+
+        Examples
+        ========
+
+        >>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
+        >>> m, n = symbols('m n')
+        >>> A = MatrixSymbol('A', n, n)
+        >>> B = MatrixSymbol('B', n, m)
+        >>> C = MatrixSymbol('C', m, n)
+        >>> D = MatrixSymbol('D', m, m)
+        >>> X = BlockMatrix([[A, B], [C, D]])
+        >>> U, D, L = X.UDLdecomposition()
+        >>> block_collapse(U*D*L)
+        Matrix([
+        [A, B],
+        [C, D]])
+
+        Raises
+        ======
+
+        ShapeError
+            If the block matrix is not a 2x2 matrix
+
+        NonInvertibleMatrixError
+            If the matrix "D" is non-invertible
+
+        See Also
+        ========
+        sympy.matrices.expressions.blockmatrix.BlockMatrix.LDUdecomposition
+        sympy.matrices.expressions.blockmatrix.BlockMatrix.LUdecomposition
+        """
+        if self.blockshape == (2,2):
+            [[A, B],
+             [C, D]] = self.blocks.tolist()
+            try:
+                DI = D.I
+            except NonInvertibleMatrixError:
+                raise NonInvertibleMatrixError('Block UDL decomposition cannot be calculated when\
+                    "D" is singular')
+            Ip = Identity(A.shape[0])
+            Iq = Identity(B.shape[1])
+            Z = ZeroMatrix(*B.shape)
+            U = BlockMatrix([[Ip, B*DI], [Z.T, Iq]])
+            D = BlockDiagMatrix(self.schur('D'), D)
+            L = BlockMatrix([[Ip, Z],[DI*C, Iq]])
+            return U, D, L
+        else:
+            raise ShapeError("Block UDL decomposition is supported only for 2x2 block matrices")
+
+    def LUdecomposition(self):
+        """Returns the Block LU decomposition of
+        a 2x2 Block Matrix
+
+        Returns
+        =======
+
+        (L, U) : Matrices
+            L : Lower Diagonal Matrix
+            U : Upper Diagonal Matrix
+
+        Examples
+        ========
+
+        >>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
+        >>> m, n = symbols('m n')
+        >>> A = MatrixSymbol('A', n, n)
+        >>> B = MatrixSymbol('B', n, m)
+        >>> C = MatrixSymbol('C', m, n)
+        >>> D = MatrixSymbol('D', m, m)
+        >>> X = BlockMatrix([[A, B], [C, D]])
+        >>> L, U = X.LUdecomposition()
+        >>> block_collapse(L*U)
+        Matrix([
+        [A, B],
+        [C, D]])
+
+        Raises
+        ======
+
+        ShapeError
+            If the block matrix is not a 2x2 matrix
+
+        NonInvertibleMatrixError
+            If the matrix "A" is non-invertible
+
+        See Also
+        ========
+        sympy.matrices.expressions.blockmatrix.BlockMatrix.UDLdecomposition
+        sympy.matrices.expressions.blockmatrix.BlockMatrix.LDUdecomposition
+        """
+        if self.blockshape == (2,2):
+            [[A, B],
+             [C, D]] = self.blocks.tolist()
+            try:
+                A = A**0.5
+                AI = A.I
+            except NonInvertibleMatrixError:
+                raise NonInvertibleMatrixError('Block LU decomposition cannot be calculated when\
+                    "A" is singular')
+            Z = ZeroMatrix(*B.shape)
+            Q = self.schur()**0.5
+            L = BlockMatrix([[A, Z], [C*AI, Q]])
+            U = BlockMatrix([[A, AI*B],[Z.T, Q]])
+            return L, U
+        else:
+            raise ShapeError("Block LU decomposition is supported only for 2x2 block matrices")
+
+    def _entry(self, i, j, **kwargs):
+        # Find row entry
+        orig_i, orig_j = i, j
+        for row_block, numrows in enumerate(self.rowblocksizes):
+            cmp = i < numrows
+            if cmp == True:
+                break
+            elif cmp == False:
+                i -= numrows
+            elif row_block < self.blockshape[0] - 1:
+                # Can't tell which block and it's not the last one, return unevaluated
+                return MatrixElement(self, orig_i, orig_j)
+        for col_block, numcols in enumerate(self.colblocksizes):
+            cmp = j < numcols
+            if cmp == True:
+                break
+            elif cmp == False:
+                j -= numcols
+            elif col_block < self.blockshape[1] - 1:
+                return MatrixElement(self, orig_i, orig_j)
+        return self.blocks[row_block, col_block][i, j]
+
+    @property
+    def is_Identity(self):
+        if self.blockshape[0] != self.blockshape[1]:
+            return False
+        for i in range(self.blockshape[0]):
+            for j in range(self.blockshape[1]):
+                if i==j and not self.blocks[i, j].is_Identity:
+                    return False
+                if i!=j and not self.blocks[i, j].is_ZeroMatrix:
+                    return False
+        return True
+
+    @property
+    def is_structurally_symmetric(self):
+        return self.rowblocksizes == self.colblocksizes
+
+    def equals(self, other):
+        if self == other:
+            return True
+        if (isinstance(other, BlockMatrix) and self.blocks == other.blocks):
+            return True
+        return super().equals(other)
+
+
+class BlockDiagMatrix(BlockMatrix):
+    """A sparse matrix with block matrices along its diagonals
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols
+    >>> n, m, l = symbols('n m l')
+    >>> X = MatrixSymbol('X', n, n)
+    >>> Y = MatrixSymbol('Y', m, m)
+    >>> BlockDiagMatrix(X, Y)
+    Matrix([
+    [X, 0],
+    [0, Y]])
+
+    Notes
+    =====
+
+    If you want to get the individual diagonal blocks, use
+    :meth:`get_diag_blocks`.
+
+    See Also
+    ========
+
+    sympy.matrices.dense.diag
+    """
+    def __new__(cls, *mats):
+        return Basic.__new__(BlockDiagMatrix, *[_sympify(m) for m in mats])
+
+    @property
+    def diag(self):
+        return self.args
+
+    @property
+    def blocks(self):
+        from sympy.matrices.immutable import ImmutableDenseMatrix
+        mats = self.args
+        data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols)
+                        for j in range(len(mats))]
+                        for i in range(len(mats))]
+        return ImmutableDenseMatrix(data, evaluate=False)
+
+    @property
+    def shape(self):
+        return (sum(block.rows for block in self.args),
+                sum(block.cols for block in self.args))
+
+    @property
+    def blockshape(self):
+        n = len(self.args)
+        return (n, n)
+
+    @property
+    def rowblocksizes(self):
+        return [block.rows for block in self.args]
+
+    @property
+    def colblocksizes(self):
+        return [block.cols for block in self.args]
+
+    def _all_square_blocks(self):
+        """Returns true if all blocks are square"""
+        return all(mat.is_square for mat in self.args)
+
+    def _eval_determinant(self):
+        if self._all_square_blocks():
+            return Mul(*[det(mat) for mat in self.args])
+        # At least one block is non-square.  Since the entire matrix must be square we know there must
+        # be at least two blocks in this matrix, in which case the entire matrix is necessarily rank-deficient
+        return S.Zero
+
+    def _eval_inverse(self, expand='ignored'):
+        if self._all_square_blocks():
+            return BlockDiagMatrix(*[mat.inverse() for mat in self.args])
+        # See comment in _eval_determinant()
+        raise NonInvertibleMatrixError('Matrix det == 0; not invertible.')
+
+    def _eval_transpose(self):
+        return BlockDiagMatrix(*[mat.transpose() for mat in self.args])
+
+    def _blockmul(self, other):
+        if (isinstance(other, BlockDiagMatrix) and
+                self.colblocksizes == other.rowblocksizes):
+            return BlockDiagMatrix(*[a*b for a, b in zip(self.args, other.args)])
+        else:
+            return BlockMatrix._blockmul(self, other)
+
+    def _blockadd(self, other):
+        if (isinstance(other, BlockDiagMatrix) and
+                self.blockshape == other.blockshape and
+                self.rowblocksizes == other.rowblocksizes and
+                self.colblocksizes == other.colblocksizes):
+            return BlockDiagMatrix(*[a + b for a, b in zip(self.args, other.args)])
+        else:
+            return BlockMatrix._blockadd(self, other)
+
+    def get_diag_blocks(self):
+        """Return the list of diagonal blocks of the matrix.
+
+        Examples
+        ========
+
+        >>> from sympy import BlockDiagMatrix, Matrix
+
+        >>> A = Matrix([[1, 2], [3, 4]])
+        >>> B = Matrix([[5, 6], [7, 8]])
+        >>> M = BlockDiagMatrix(A, B)
+
+        How to get diagonal blocks from the block diagonal matrix:
+
+        >>> diag_blocks = M.get_diag_blocks()
+        >>> diag_blocks[0]
+        Matrix([
+        [1, 2],
+        [3, 4]])
+        >>> diag_blocks[1]
+        Matrix([
+        [5, 6],
+        [7, 8]])
+        """
+        return self.args
+
+
+def block_collapse(expr):
+    """Evaluates a block matrix expression
+
+    >>> from sympy import MatrixSymbol, BlockMatrix, symbols, Identity, ZeroMatrix, block_collapse
+    >>> n,m,l = symbols('n m l')
+    >>> X = MatrixSymbol('X', n, n)
+    >>> Y = MatrixSymbol('Y', m, m)
+    >>> Z = MatrixSymbol('Z', n, m)
+    >>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
+    >>> print(B)
+    Matrix([
+    [X, Z],
+    [0, Y]])
+
+    >>> C = BlockMatrix([[Identity(n), Z]])
+    >>> print(C)
+    Matrix([[I, Z]])
+
+    >>> print(block_collapse(C*B))
+    Matrix([[X, Z + Z*Y]])
+    """
+    from sympy.strategies.util import expr_fns
+
+    hasbm = lambda expr: isinstance(expr, MatrixExpr) and expr.has(BlockMatrix)
+
+    conditioned_rl = condition(
+        hasbm,
+        typed(
+            {MatAdd: do_one(bc_matadd, bc_block_plus_ident),
+             MatMul: do_one(bc_matmul, bc_dist),
+             MatPow: bc_matmul,
+             Transpose: bc_transpose,
+             Inverse: bc_inverse,
+             BlockMatrix: do_one(bc_unpack, deblock)}
+        )
+    )
+
+    rule = exhaust(
+        bottom_up(
+            exhaust(conditioned_rl),
+            fns=expr_fns
+        )
+    )
+
+    result = rule(expr)
+    doit = getattr(result, 'doit', None)
+    if doit is not None:
+        return doit()
+    else:
+        return result
+
+def bc_unpack(expr):
+    if expr.blockshape == (1, 1):
+        return expr.blocks[0, 0]
+    return expr
+
+def bc_matadd(expr):
+    args = sift(expr.args, lambda M: isinstance(M, BlockMatrix))
+    blocks = args[True]
+    if not blocks:
+        return expr
+
+    nonblocks = args[False]
+    block = blocks[0]
+    for b in blocks[1:]:
+        block = block._blockadd(b)
+    if nonblocks:
+        return MatAdd(*nonblocks) + block
+    else:
+        return block
+
+def bc_block_plus_ident(expr):
+    idents = [arg for arg in expr.args if arg.is_Identity]
+    if not idents:
+        return expr
+
+    blocks = [arg for arg in expr.args if isinstance(arg, BlockMatrix)]
+    if (blocks and all(b.structurally_equal(blocks[0]) for b in blocks)
+               and blocks[0].is_structurally_symmetric):
+        block_id = BlockDiagMatrix(*[Identity(k)
+                                        for k in blocks[0].rowblocksizes])
+        rest = [arg for arg in expr.args if not arg.is_Identity and not isinstance(arg, BlockMatrix)]
+        return MatAdd(block_id * len(idents), *blocks, *rest).doit()
+
+    return expr
+
+def bc_dist(expr):
+    """ Turn  a*[X, Y] into [a*X, a*Y] """
+    factor, mat = expr.as_coeff_mmul()
+    if factor == 1:
+        return expr
+
+    unpacked = unpack(mat)
+
+    if isinstance(unpacked, BlockDiagMatrix):
+        B = unpacked.diag
+        new_B = [factor * mat for mat in B]
+        return BlockDiagMatrix(*new_B)
+    elif isinstance(unpacked, BlockMatrix):
+        B = unpacked.blocks
+        new_B = [
+            [factor * B[i, j] for j in range(B.cols)] for i in range(B.rows)]
+        return BlockMatrix(new_B)
+    return expr
+
+
+def bc_matmul(expr):
+    if isinstance(expr, MatPow):
+        if expr.args[1].is_Integer:
+            factor, matrices = (1, [expr.args[0]]*expr.args[1])
+        else:
+            return expr
+    else:
+        factor, matrices = expr.as_coeff_matrices()
+
+    i = 0
+    while (i+1 < len(matrices)):
+        A, B = matrices[i:i+2]
+        if isinstance(A, BlockMatrix) and isinstance(B, BlockMatrix):
+            matrices[i] = A._blockmul(B)
+            matrices.pop(i+1)
+        elif isinstance(A, BlockMatrix):
+            matrices[i] = A._blockmul(BlockMatrix([[B]]))
+            matrices.pop(i+1)
+        elif isinstance(B, BlockMatrix):
+            matrices[i] = BlockMatrix([[A]])._blockmul(B)
+            matrices.pop(i+1)
+        else:
+            i+=1
+    return MatMul(factor, *matrices).doit()
+
+def bc_transpose(expr):
+    collapse = block_collapse(expr.arg)
+    return collapse._eval_transpose()
+
+
+def bc_inverse(expr):
+    if isinstance(expr.arg, BlockDiagMatrix):
+        return expr.inverse()
+
+    expr2 = blockinverse_1x1(expr)
+    if expr != expr2:
+        return expr2
+    return blockinverse_2x2(Inverse(reblock_2x2(expr.arg)))
+
+def blockinverse_1x1(expr):
+    if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (1, 1):
+        mat = Matrix([[expr.arg.blocks[0].inverse()]])
+        return BlockMatrix(mat)
+    return expr
+
+
+def blockinverse_2x2(expr):
+    if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (2, 2):
+        # See: Inverses of 2x2 Block Matrices, Tzon-Tzer Lu and Sheng-Hua Shiou
+        [[A, B],
+         [C, D]] = expr.arg.blocks.tolist()
+
+        formula = _choose_2x2_inversion_formula(A, B, C, D)
+        if formula != None:
+            MI = expr.arg.schur(formula).I
+        if formula == 'A':
+            AI = A.I
+            return BlockMatrix([[AI + AI * B * MI * C * AI, -AI * B * MI], [-MI * C * AI, MI]])
+        if formula == 'B':
+            BI = B.I
+            return BlockMatrix([[-MI * D * BI, MI], [BI + BI * A * MI * D * BI, -BI * A * MI]])
+        if formula == 'C':
+            CI = C.I
+            return BlockMatrix([[-CI * D * MI, CI + CI * D * MI * A * CI], [MI, -MI * A * CI]])
+        if formula == 'D':
+            DI = D.I
+            return BlockMatrix([[MI, -MI * B * DI], [-DI * C * MI, DI + DI * C * MI * B * DI]])
+
+    return expr
+
+
+def _choose_2x2_inversion_formula(A, B, C, D):
+    """
+    Assuming [[A, B], [C, D]] would form a valid square block matrix, find
+    which of the classical 2x2 block matrix inversion formulas would be
+    best suited.
+
+    Returns 'A', 'B', 'C', 'D' to represent the algorithm involving inversion
+    of the given argument or None if the matrix cannot be inverted using
+    any of those formulas.
+    """
+    # Try to find a known invertible matrix.  Note that the Schur complement
+    # is currently not being considered for this
+    A_inv = ask(Q.invertible(A))
+    if A_inv == True:
+        return 'A'
+    B_inv = ask(Q.invertible(B))
+    if B_inv == True:
+        return 'B'
+    C_inv = ask(Q.invertible(C))
+    if C_inv == True:
+        return 'C'
+    D_inv = ask(Q.invertible(D))
+    if D_inv == True:
+        return 'D'
+    # Otherwise try to find a matrix that isn't known to be non-invertible
+    if A_inv != False:
+        return 'A'
+    if B_inv != False:
+        return 'B'
+    if C_inv != False:
+        return 'C'
+    if D_inv != False:
+        return 'D'
+    return None
+
+
+def deblock(B):
+    """ Flatten a BlockMatrix of BlockMatrices """
+    if not isinstance(B, BlockMatrix) or not B.blocks.has(BlockMatrix):
+        return B
+    wrap = lambda x: x if isinstance(x, BlockMatrix) else BlockMatrix([[x]])
+    bb = B.blocks.applyfunc(wrap)  # everything is a block
+
+    try:
+        MM = Matrix(0, sum(bb[0, i].blocks.shape[1] for i in range(bb.shape[1])), [])
+        for row in range(0, bb.shape[0]):
+            M = Matrix(bb[row, 0].blocks)
+            for col in range(1, bb.shape[1]):
+                M = M.row_join(bb[row, col].blocks)
+            MM = MM.col_join(M)
+
+        return BlockMatrix(MM)
+    except ShapeError:
+        return B
+
+
+def reblock_2x2(expr):
+    """
+    Reblock a BlockMatrix so that it has 2x2 blocks of block matrices.  If
+    possible in such a way that the matrix continues to be invertible using the
+    classical 2x2 block inversion formulas.
+    """
+    if not isinstance(expr, BlockMatrix) or not all(d > 2 for d in expr.blockshape):
+        return expr
+
+    BM = BlockMatrix  # for brevity's sake
+    rowblocks, colblocks = expr.blockshape
+    blocks = expr.blocks
+    for i in range(1, rowblocks):
+        for j in range(1, colblocks):
+            # try to split rows at i and cols at j
+            A = bc_unpack(BM(blocks[:i, :j]))
+            B = bc_unpack(BM(blocks[:i, j:]))
+            C = bc_unpack(BM(blocks[i:, :j]))
+            D = bc_unpack(BM(blocks[i:, j:]))
+
+            formula = _choose_2x2_inversion_formula(A, B, C, D)
+            if formula is not None:
+                return BlockMatrix([[A, B], [C, D]])
+
+    # else: nothing worked, just split upper left corner
+    return BM([[blocks[0, 0], BM(blocks[0, 1:])],
+               [BM(blocks[1:, 0]), BM(blocks[1:, 1:])]])
+
+
+def bounds(sizes):
+    """ Convert sequence of numbers into pairs of low-high pairs
+
+    >>> from sympy.matrices.expressions.blockmatrix import bounds
+    >>> bounds((1, 10, 50))
+    [(0, 1), (1, 11), (11, 61)]
+    """
+    low = 0
+    rv = []
+    for size in sizes:
+        rv.append((low, low + size))
+        low += size
+    return rv
+
+def blockcut(expr, rowsizes, colsizes):
+    """ Cut a matrix expression into Blocks
+
+    >>> from sympy import ImmutableMatrix, blockcut
+    >>> M = ImmutableMatrix(4, 4, range(16))
+    >>> B = blockcut(M, (1, 3), (1, 3))
+    >>> type(B).__name__
+    'BlockMatrix'
+    >>> ImmutableMatrix(B.blocks[0, 1])
+    Matrix([[1, 2, 3]])
+    """
+
+    rowbounds = bounds(rowsizes)
+    colbounds = bounds(colsizes)
+    return BlockMatrix([[MatrixSlice(expr, rowbound, colbound)
+                         for colbound in colbounds]
+                         for rowbound in rowbounds])

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/companion.py

@@ -0,0 +1,56 @@
+from sympy.core.singleton import S
+from sympy.core.sympify import _sympify
+from sympy.polys.polytools import Poly
+
+from .matexpr import MatrixExpr
+
+
+class CompanionMatrix(MatrixExpr):
+    """A symbolic companion matrix of a polynomial.
+
+    Examples
+    ========
+
+    >>> from sympy import Poly, Symbol, symbols
+    >>> from sympy.matrices.expressions import CompanionMatrix
+    >>> x = Symbol('x')
+    >>> c0, c1, c2, c3, c4 = symbols('c0:5')
+    >>> p = Poly(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + x**5, x)
+    >>> CompanionMatrix(p)
+    CompanionMatrix(Poly(x**5 + c4*x**4 + c3*x**3 + c2*x**2 + c1*x + c0,
+    x, domain='ZZ[c0,c1,c2,c3,c4]'))
+    """
+    def __new__(cls, poly):
+        poly = _sympify(poly)
+        if not isinstance(poly, Poly):
+            raise ValueError("{} must be a Poly instance.".format(poly))
+        if not poly.is_monic:
+            raise ValueError("{} must be a monic polynomial.".format(poly))
+        if not poly.is_univariate:
+            raise ValueError(
+                "{} must be a univariate polynomial.".format(poly))
+        if not poly.degree() >= 1:
+            raise ValueError(
+                "{} must have degree not less than 1.".format(poly))
+
+        return super().__new__(cls, poly)
+
+
+    @property
+    def shape(self):
+        poly = self.args[0]
+        size = poly.degree()
+        return size, size
+
+
+    def _entry(self, i, j):
+        if j == self.cols - 1:
+            return -self.args[0].all_coeffs()[-1 - i]
+        elif i == j + 1:
+            return S.One
+        return S.Zero
+
+
+    def as_explicit(self):
+        from sympy.matrices.immutable import ImmutableDenseMatrix
+        return ImmutableDenseMatrix.companion(self.args[0])

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/determinant.py

@@ -0,0 +1,141 @@
+from sympy.core.basic import Basic
+from sympy.core.expr import Expr
+from sympy.core.singleton import S
+from sympy.core.sympify import sympify
+from sympy.matrices.common import NonSquareMatrixError
+
+
+class Determinant(Expr):
+    """Matrix Determinant
+
+    Represents the determinant of a matrix expression.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, Determinant, eye
+    >>> A = MatrixSymbol('A', 3, 3)
+    >>> Determinant(A)
+    Determinant(A)
+    >>> Determinant(eye(3)).doit()
+    1
+    """
+    is_commutative = True
+
+    def __new__(cls, mat):
+        mat = sympify(mat)
+        if not mat.is_Matrix:
+            raise TypeError("Input to Determinant, %s, not a matrix" % str(mat))
+
+        if mat.is_square is False:
+            raise NonSquareMatrixError("Det of a non-square matrix")
+
+        return Basic.__new__(cls, mat)
+
+    @property
+    def arg(self):
+        return self.args[0]
+
+    @property
+    def kind(self):
+        return self.arg.kind.element_kind
+
+    def doit(self, expand=False, **hints):
+        try:
+            return self.arg._eval_determinant()
+        except (AttributeError, NotImplementedError):
+            return self
+
+def det(matexpr):
+    """ Matrix Determinant
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, det, eye
+    >>> A = MatrixSymbol('A', 3, 3)
+    >>> det(A)
+    Determinant(A)
+    >>> det(eye(3))
+    1
+    """
+
+    return Determinant(matexpr).doit()
+
+class Permanent(Expr):
+    """Matrix Permanent
+
+    Represents the permanent of a matrix expression.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, Permanent, ones
+    >>> A = MatrixSymbol('A', 3, 3)
+    >>> Permanent(A)
+    Permanent(A)
+    >>> Permanent(ones(3, 3)).doit()
+    6
+    """
+
+    def __new__(cls, mat):
+        mat = sympify(mat)
+        if not mat.is_Matrix:
+            raise TypeError("Input to Permanent, %s, not a matrix" % str(mat))
+
+        return Basic.__new__(cls, mat)
+
+    @property
+    def arg(self):
+        return self.args[0]
+
+    def doit(self, expand=False, **hints):
+        try:
+            return self.arg.per()
+        except (AttributeError, NotImplementedError):
+            return self
+
+def per(matexpr):
+    """ Matrix Permanent
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, Matrix, per, ones
+    >>> A = MatrixSymbol('A', 3, 3)
+    >>> per(A)
+    Permanent(A)
+    >>> per(ones(5, 5))
+    120
+    >>> M = Matrix([1, 2, 5])
+    >>> per(M)
+    8
+    """
+
+    return Permanent(matexpr).doit()
+
+from sympy.assumptions.ask import ask, Q
+from sympy.assumptions.refine import handlers_dict
+
+
+def refine_Determinant(expr, assumptions):
+    """
+    >>> from sympy import MatrixSymbol, Q, assuming, refine, det
+    >>> X = MatrixSymbol('X', 2, 2)
+    >>> det(X)
+    Determinant(X)
+    >>> with assuming(Q.orthogonal(X)):
+    ...     print(refine(det(X)))
+    1
+    """
+    if ask(Q.orthogonal(expr.arg), assumptions):
+        return S.One
+    elif ask(Q.singular(expr.arg), assumptions):
+        return S.Zero
+    elif ask(Q.unit_triangular(expr.arg), assumptions):
+        return S.One
+
+    return expr
+
+
+handlers_dict['Determinant'] = refine_Determinant

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/diagonal.py

@@ -0,0 +1,220 @@
+from sympy.core.sympify import _sympify
+
+from sympy.matrices.expressions import MatrixExpr
+from sympy.core import S, Eq, Ge
+from sympy.core.mul import Mul
+from sympy.functions.special.tensor_functions import KroneckerDelta
+
+
+class DiagonalMatrix(MatrixExpr):
+    """DiagonalMatrix(M) will create a matrix expression that
+    behaves as though all off-diagonal elements,
+    `M[i, j]` where `i != j`, are zero.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, DiagonalMatrix, Symbol
+    >>> n = Symbol('n', integer=True)
+    >>> m = Symbol('m', integer=True)
+    >>> D = DiagonalMatrix(MatrixSymbol('x', 2, 3))
+    >>> D[1, 2]
+    0
+    >>> D[1, 1]
+    x[1, 1]
+
+    The length of the diagonal -- the lesser of the two dimensions of `M` --
+    is accessed through the `diagonal_length` property:
+
+    >>> D.diagonal_length
+    2
+    >>> DiagonalMatrix(MatrixSymbol('x', n + 1, n)).diagonal_length
+    n
+
+    When one of the dimensions is symbolic the other will be treated as
+    though it is smaller:
+
+    >>> tall = DiagonalMatrix(MatrixSymbol('x', n, 3))
+    >>> tall.diagonal_length
+    3
+    >>> tall[10, 1]
+    0
+
+    When the size of the diagonal is not known, a value of None will
+    be returned:
+
+    >>> DiagonalMatrix(MatrixSymbol('x', n, m)).diagonal_length is None
+    True
+
+    """
+    arg = property(lambda self: self.args[0])
+
+    shape = property(lambda self: self.arg.shape)  # type:ignore
+
+    @property
+    def diagonal_length(self):
+        r, c = self.shape
+        if r.is_Integer and c.is_Integer:
+            m = min(r, c)
+        elif r.is_Integer and not c.is_Integer:
+            m = r
+        elif c.is_Integer and not r.is_Integer:
+            m = c
+        elif r == c:
+            m = r
+        else:
+            try:
+                m = min(r, c)
+            except TypeError:
+                m = None
+        return m
+
+    def _entry(self, i, j, **kwargs):
+        if self.diagonal_length is not None:
+            if Ge(i, self.diagonal_length) is S.true:
+                return S.Zero
+            elif Ge(j, self.diagonal_length) is S.true:
+                return S.Zero
+        eq = Eq(i, j)
+        if eq is S.true:
+            return self.arg[i, i]
+        elif eq is S.false:
+            return S.Zero
+        return self.arg[i, j]*KroneckerDelta(i, j)
+
+
+class DiagonalOf(MatrixExpr):
+    """DiagonalOf(M) will create a matrix expression that
+    is equivalent to the diagonal of `M`, represented as
+    a single column matrix.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, DiagonalOf, Symbol
+    >>> n = Symbol('n', integer=True)
+    >>> m = Symbol('m', integer=True)
+    >>> x = MatrixSymbol('x', 2, 3)
+    >>> diag = DiagonalOf(x)
+    >>> diag.shape
+    (2, 1)
+
+    The diagonal can be addressed like a matrix or vector and will
+    return the corresponding element of the original matrix:
+
+    >>> diag[1, 0] == diag[1] == x[1, 1]
+    True
+
+    The length of the diagonal -- the lesser of the two dimensions of `M` --
+    is accessed through the `diagonal_length` property:
+
+    >>> diag.diagonal_length
+    2
+    >>> DiagonalOf(MatrixSymbol('x', n + 1, n)).diagonal_length
+    n
+
+    When only one of the dimensions is symbolic the other will be
+    treated as though it is smaller:
+
+    >>> dtall = DiagonalOf(MatrixSymbol('x', n, 3))
+    >>> dtall.diagonal_length
+    3
+
+    When the size of the diagonal is not known, a value of None will
+    be returned:
+
+    >>> DiagonalOf(MatrixSymbol('x', n, m)).diagonal_length is None
+    True
+
+    """
+    arg = property(lambda self: self.args[0])
+    @property
+    def shape(self):
+        r, c = self.arg.shape
+        if r.is_Integer and c.is_Integer:
+            m = min(r, c)
+        elif r.is_Integer and not c.is_Integer:
+            m = r
+        elif c.is_Integer and not r.is_Integer:
+            m = c
+        elif r == c:
+            m = r
+        else:
+            try:
+                m = min(r, c)
+            except TypeError:
+                m = None
+        return m, S.One
+
+    @property
+    def diagonal_length(self):
+        return self.shape[0]
+
+    def _entry(self, i, j, **kwargs):
+        return self.arg._entry(i, i, **kwargs)
+
+
+class DiagMatrix(MatrixExpr):
+    """
+    Turn a vector into a diagonal matrix.
+    """
+    def __new__(cls, vector):
+        vector = _sympify(vector)
+        obj = MatrixExpr.__new__(cls, vector)
+        shape = vector.shape
+        dim = shape[1] if shape[0] == 1 else shape[0]
+        if vector.shape[0] != 1:
+            obj._iscolumn = True
+        else:
+            obj._iscolumn = False
+        obj._shape = (dim, dim)
+        obj._vector = vector
+        return obj
+
+    @property
+    def shape(self):
+        return self._shape
+
+    def _entry(self, i, j, **kwargs):
+        if self._iscolumn:
+            result = self._vector._entry(i, 0, **kwargs)
+        else:
+            result = self._vector._entry(0, j, **kwargs)
+        if i != j:
+            result *= KroneckerDelta(i, j)
+        return result
+
+    def _eval_transpose(self):
+        return self
+
+    def as_explicit(self):
+        from sympy.matrices.dense import diag
+        return diag(*list(self._vector.as_explicit()))
+
+    def doit(self, **hints):
+        from sympy.assumptions import ask, Q
+        from sympy.matrices.expressions.matmul import MatMul
+        from sympy.matrices.expressions.transpose import Transpose
+        from sympy.matrices.dense import eye
+        from sympy.matrices.matrices import MatrixBase
+        vector = self._vector
+        # This accounts for shape (1, 1) and identity matrices, among others:
+        if ask(Q.diagonal(vector)):
+            return vector
+        if isinstance(vector, MatrixBase):
+            ret = eye(max(vector.shape))
+            for i in range(ret.shape[0]):
+                ret[i, i] = vector[i]
+            return type(vector)(ret)
+        if vector.is_MatMul:
+            matrices = [arg for arg in vector.args if arg.is_Matrix]
+            scalars = [arg for arg in vector.args if arg not in matrices]
+            if scalars:
+                return Mul.fromiter(scalars)*DiagMatrix(MatMul.fromiter(matrices).doit()).doit()
+        if isinstance(vector, Transpose):
+            vector = vector.arg
+        return DiagMatrix(vector)
+
+
+def diagonalize_vector(vector):
+    return DiagMatrix(vector).doit()

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/dotproduct.py

@@ -0,0 +1,55 @@
+from sympy.core import Basic, Expr
+from sympy.core.sympify import _sympify
+from sympy.matrices.expressions.transpose import transpose
+
+
+class DotProduct(Expr):
+    """
+    Dot product of vector matrices
+
+    The input should be two 1 x n or n x 1 matrices. The output represents the
+    scalar dotproduct.
+
+    This is similar to using MatrixElement and MatMul, except DotProduct does
+    not require that one vector to be a row vector and the other vector to be
+    a column vector.
+
+    >>> from sympy import MatrixSymbol, DotProduct
+    >>> A = MatrixSymbol('A', 1, 3)
+    >>> B = MatrixSymbol('B', 1, 3)
+    >>> DotProduct(A, B)
+    DotProduct(A, B)
+    >>> DotProduct(A, B).doit()
+    A[0, 0]*B[0, 0] + A[0, 1]*B[0, 1] + A[0, 2]*B[0, 2]
+    """
+
+    def __new__(cls, arg1, arg2):
+        arg1, arg2 = _sympify((arg1, arg2))
+
+        if not arg1.is_Matrix:
+            raise TypeError("Argument 1 of DotProduct is not a matrix")
+        if not arg2.is_Matrix:
+            raise TypeError("Argument 2 of DotProduct is not a matrix")
+        if not (1 in arg1.shape):
+            raise TypeError("Argument 1 of DotProduct is not a vector")
+        if not (1 in arg2.shape):
+            raise TypeError("Argument 2 of DotProduct is not a vector")
+
+        if set(arg1.shape) != set(arg2.shape):
+            raise TypeError("DotProduct arguments are not the same length")
+
+        return Basic.__new__(cls, arg1, arg2)
+
+    def doit(self, expand=False, **hints):
+        if self.args[0].shape == self.args[1].shape:
+            if self.args[0].shape[0] == 1:
+                mul = self.args[0]*transpose(self.args[1])
+            else:
+                mul = transpose(self.args[0])*self.args[1]
+        else:
+            if self.args[0].shape[0] == 1:
+                mul = self.args[0]*self.args[1]
+            else:
+                mul = transpose(self.args[0])*transpose(self.args[1])
+
+        return mul[0]

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/factorizations.py

@@ -0,0 +1,62 @@
+from sympy.matrices.expressions import MatrixExpr
+from sympy.assumptions.ask import Q
+
+class Factorization(MatrixExpr):
+    arg = property(lambda self: self.args[0])
+    shape = property(lambda self: self.arg.shape)  # type: ignore
+
+class LofLU(Factorization):
+    @property
+    def predicates(self):
+        return (Q.lower_triangular,)
+class UofLU(Factorization):
+    @property
+    def predicates(self):
+        return (Q.upper_triangular,)
+
+class LofCholesky(LofLU): pass
+class UofCholesky(UofLU): pass
+
+class QofQR(Factorization):
+    @property
+    def predicates(self):
+        return (Q.orthogonal,)
+class RofQR(Factorization):
+    @property
+    def predicates(self):
+        return (Q.upper_triangular,)
+
+class EigenVectors(Factorization):
+    @property
+    def predicates(self):
+        return (Q.orthogonal,)
+class EigenValues(Factorization):
+    @property
+    def predicates(self):
+        return (Q.diagonal,)
+
+class UofSVD(Factorization):
+    @property
+    def predicates(self):
+        return (Q.orthogonal,)
+class SofSVD(Factorization):
+    @property
+    def predicates(self):
+        return (Q.diagonal,)
+class VofSVD(Factorization):
+    @property
+    def predicates(self):
+        return (Q.orthogonal,)
+
+
+def lu(expr):
+    return LofLU(expr), UofLU(expr)
+
+def qr(expr):
+    return QofQR(expr), RofQR(expr)
+
+def eig(expr):
+    return EigenValues(expr), EigenVectors(expr)
+
+def svd(expr):
+    return UofSVD(expr), SofSVD(expr), VofSVD(expr)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/fourier.py

@@ -0,0 +1,91 @@
+from sympy.core.sympify import _sympify
+from sympy.matrices.expressions import MatrixExpr
+from sympy.core.numbers import I
+from sympy.core.singleton import S
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+
+
+class DFT(MatrixExpr):
+    r"""
+    Returns a discrete Fourier transform matrix. The matrix is scaled
+    with :math:`\frac{1}{\sqrt{n}}` so that it is unitary.
+
+    Parameters
+    ==========
+
+    n : integer or Symbol
+        Size of the transform.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import n
+    >>> from sympy.matrices.expressions.fourier import DFT
+    >>> DFT(3)
+    DFT(3)
+    >>> DFT(3).as_explicit()
+    Matrix([
+    [sqrt(3)/3,                sqrt(3)/3,                sqrt(3)/3],
+    [sqrt(3)/3, sqrt(3)*exp(-2*I*pi/3)/3,  sqrt(3)*exp(2*I*pi/3)/3],
+    [sqrt(3)/3,  sqrt(3)*exp(2*I*pi/3)/3, sqrt(3)*exp(-2*I*pi/3)/3]])
+    >>> DFT(n).shape
+    (n, n)
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/DFT_matrix
+
+    """
+
+    def __new__(cls, n):
+        n = _sympify(n)
+        cls._check_dim(n)
+
+        obj = super().__new__(cls, n)
+        return obj
+
+    n = property(lambda self: self.args[0])  # type: ignore
+    shape = property(lambda self: (self.n, self.n))  # type: ignore
+
+    def _entry(self, i, j, **kwargs):
+        w = exp(-2*S.Pi*I/self.n)
+        return w**(i*j) / sqrt(self.n)
+
+    def _eval_inverse(self):
+        return IDFT(self.n)
+
+
+class IDFT(DFT):
+    r"""
+    Returns an inverse discrete Fourier transform matrix. The matrix is scaled
+    with :math:`\frac{1}{\sqrt{n}}` so that it is unitary.
+
+    Parameters
+    ==========
+
+    n : integer or Symbol
+        Size of the transform
+
+    Examples
+    ========
+
+    >>> from sympy.matrices.expressions.fourier import DFT, IDFT
+    >>> IDFT(3)
+    IDFT(3)
+    >>> IDFT(4)*DFT(4)
+    I
+
+    See Also
+    ========
+
+    DFT
+
+    """
+    def _entry(self, i, j, **kwargs):
+        w = exp(-2*S.Pi*I/self.n)
+        return w**(-i*j) / sqrt(self.n)
+
+    def _eval_inverse(self):
+        return DFT(self.n)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/funcmatrix.py

@@ -0,0 +1,118 @@
+from .matexpr import MatrixExpr
+from sympy.core.function import FunctionClass, Lambda
+from sympy.core.symbol import Dummy
+from sympy.core.sympify import _sympify, sympify
+from sympy.matrices import Matrix
+from sympy.functions.elementary.complexes import re, im
+
+
+class FunctionMatrix(MatrixExpr):
+    """Represents a matrix using a function (``Lambda``) which gives
+    outputs according to the coordinates of each matrix entries.
+
+    Parameters
+    ==========
+
+    rows : nonnegative integer. Can be symbolic.
+
+    cols : nonnegative integer. Can be symbolic.
+
+    lamda : Function, Lambda or str
+        If it is a SymPy ``Function`` or ``Lambda`` instance,
+        it should be able to accept two arguments which represents the
+        matrix coordinates.
+
+        If it is a pure string containing Python ``lambda`` semantics,
+        it is interpreted by the SymPy parser and casted into a SymPy
+        ``Lambda`` instance.
+
+    Examples
+    ========
+
+    Creating a ``FunctionMatrix`` from ``Lambda``:
+
+    >>> from sympy import FunctionMatrix, symbols, Lambda, MatPow
+    >>> i, j, n, m = symbols('i,j,n,m')
+    >>> FunctionMatrix(n, m, Lambda((i, j), i + j))
+    FunctionMatrix(n, m, Lambda((i, j), i + j))
+
+    Creating a ``FunctionMatrix`` from a SymPy function:
+
+    >>> from sympy import KroneckerDelta
+    >>> X = FunctionMatrix(3, 3, KroneckerDelta)
+    >>> X.as_explicit()
+    Matrix([
+    [1, 0, 0],
+    [0, 1, 0],
+    [0, 0, 1]])
+
+    Creating a ``FunctionMatrix`` from a SymPy undefined function:
+
+    >>> from sympy import Function
+    >>> f = Function('f')
+    >>> X = FunctionMatrix(3, 3, f)
+    >>> X.as_explicit()
+    Matrix([
+    [f(0, 0), f(0, 1), f(0, 2)],
+    [f(1, 0), f(1, 1), f(1, 2)],
+    [f(2, 0), f(2, 1), f(2, 2)]])
+
+    Creating a ``FunctionMatrix`` from Python ``lambda``:
+
+    >>> FunctionMatrix(n, m, 'lambda i, j: i + j')
+    FunctionMatrix(n, m, Lambda((i, j), i + j))
+
+    Example of lazy evaluation of matrix product:
+
+    >>> Y = FunctionMatrix(1000, 1000, Lambda((i, j), i + j))
+    >>> isinstance(Y*Y, MatPow) # this is an expression object
+    True
+    >>> (Y**2)[10,10] # So this is evaluated lazily
+    342923500
+
+    Notes
+    =====
+
+    This class provides an alternative way to represent an extremely
+    dense matrix with entries in some form of a sequence, in a most
+    sparse way.
+    """
+    def __new__(cls, rows, cols, lamda):
+        rows, cols = _sympify(rows), _sympify(cols)
+        cls._check_dim(rows)
+        cls._check_dim(cols)
+
+        lamda = sympify(lamda)
+        if not isinstance(lamda, (FunctionClass, Lambda)):
+            raise ValueError(
+                "{} should be compatible with SymPy function classes."
+                .format(lamda))
+
+        if 2 not in lamda.nargs:
+            raise ValueError(
+                '{} should be able to accept 2 arguments.'.format(lamda))
+
+        if not isinstance(lamda, Lambda):
+            i, j = Dummy('i'), Dummy('j')
+            lamda = Lambda((i, j), lamda(i, j))
+
+        return super().__new__(cls, rows, cols, lamda)
+
+    @property
+    def shape(self):
+        return self.args[0:2]
+
+    @property
+    def lamda(self):
+        return self.args[2]
+
+    def _entry(self, i, j, **kwargs):
+        return self.lamda(i, j)
+
+    def _eval_trace(self):
+        from sympy.matrices.expressions.trace import Trace
+        from sympy.concrete.summations import Sum
+        return Trace(self).rewrite(Sum).doit()
+
+    def _eval_as_real_imag(self):
+        return (re(Matrix(self)), im(Matrix(self)))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/hadamard.py

@@ -0,0 +1,464 @@
+from collections import Counter
+
+from sympy.core import Mul, sympify
+from sympy.core.add import Add
+from sympy.core.expr import ExprBuilder
+from sympy.core.sorting import default_sort_key
+from sympy.functions.elementary.exponential import log
+from sympy.matrices.expressions.matexpr import MatrixExpr
+from sympy.matrices.expressions._shape import validate_matadd_integer as validate
+from sympy.matrices.expressions.special import ZeroMatrix, OneMatrix
+from sympy.strategies import (
+    unpack, flatten, condition, exhaust, rm_id, sort
+)
+from sympy.utilities.exceptions import sympy_deprecation_warning
+
+
+def hadamard_product(*matrices):
+    """
+    Return the elementwise (aka Hadamard) product of matrices.
+
+    Examples
+    ========
+
+    >>> from sympy import hadamard_product, MatrixSymbol
+    >>> A = MatrixSymbol('A', 2, 3)
+    >>> B = MatrixSymbol('B', 2, 3)
+    >>> hadamard_product(A)
+    A
+    >>> hadamard_product(A, B)
+    HadamardProduct(A, B)
+    >>> hadamard_product(A, B)[0, 1]
+    A[0, 1]*B[0, 1]
+    """
+    if not matrices:
+        raise TypeError("Empty Hadamard product is undefined")
+    if len(matrices) == 1:
+        return matrices[0]
+    return HadamardProduct(*matrices).doit()
+
+
+class HadamardProduct(MatrixExpr):
+    """
+    Elementwise product of matrix expressions
+
+    Examples
+    ========
+
+    Hadamard product for matrix symbols:
+
+    >>> from sympy import hadamard_product, HadamardProduct, MatrixSymbol
+    >>> A = MatrixSymbol('A', 5, 5)
+    >>> B = MatrixSymbol('B', 5, 5)
+    >>> isinstance(hadamard_product(A, B), HadamardProduct)
+    True
+
+    Notes
+    =====
+
+    This is a symbolic object that simply stores its argument without
+    evaluating it. To actually compute the product, use the function
+    ``hadamard_product()`` or ``HadamardProduct.doit``
+    """
+    is_HadamardProduct = True
+
+    def __new__(cls, *args, evaluate=False, check=None):
+        args = list(map(sympify, args))
+        if len(args) == 0:
+            # We currently don't have a way to support one-matrices of generic dimensions:
+            raise ValueError("HadamardProduct needs at least one argument")
+
+        if not all(isinstance(arg, MatrixExpr) for arg in args):
+            raise TypeError("Mix of Matrix and Scalar symbols")
+
+        if check is not None:
+            sympy_deprecation_warning(
+                "Passing check to HadamardProduct is deprecated and the check argument will be removed in a future version.",
+                deprecated_since_version="1.11",
+                active_deprecations_target='remove-check-argument-from-matrix-operations')
+
+        if check is not False:
+            validate(*args)
+
+        obj = super().__new__(cls, *args)
+        if evaluate:
+            obj = obj.doit(deep=False)
+        return obj
+
+    @property
+    def shape(self):
+        return self.args[0].shape
+
+    def _entry(self, i, j, **kwargs):
+        return Mul(*[arg._entry(i, j, **kwargs) for arg in self.args])
+
+    def _eval_transpose(self):
+        from sympy.matrices.expressions.transpose import transpose
+        return HadamardProduct(*list(map(transpose, self.args)))
+
+    def doit(self, **hints):
+        expr = self.func(*(i.doit(**hints) for i in self.args))
+        # Check for explicit matrices:
+        from sympy.matrices.matrices import MatrixBase
+        from sympy.matrices.immutable import ImmutableMatrix
+
+        explicit = [i for i in expr.args if isinstance(i, MatrixBase)]
+        if explicit:
+            remainder = [i for i in expr.args if i not in explicit]
+            expl_mat = ImmutableMatrix([
+                Mul.fromiter(i) for i in zip(*explicit)
+            ]).reshape(*self.shape)
+            expr = HadamardProduct(*([expl_mat] + remainder))
+
+        return canonicalize(expr)
+
+    def _eval_derivative(self, x):
+        terms = []
+        args = list(self.args)
+        for i in range(len(args)):
+            factors = args[:i] + [args[i].diff(x)] + args[i+1:]
+            terms.append(hadamard_product(*factors))
+        return Add.fromiter(terms)
+
+    def _eval_derivative_matrix_lines(self, x):
+        from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
+        from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
+        from sympy.matrices.expressions.matexpr import _make_matrix
+
+        with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
+        lines = []
+        for ind in with_x_ind:
+            left_args = self.args[:ind]
+            right_args = self.args[ind+1:]
+
+            d = self.args[ind]._eval_derivative_matrix_lines(x)
+            hadam = hadamard_product(*(right_args + left_args))
+            diagonal = [(0, 2), (3, 4)]
+            diagonal = [e for j, e in enumerate(diagonal) if self.shape[j] != 1]
+            for i in d:
+                l1 = i._lines[i._first_line_index]
+                l2 = i._lines[i._second_line_index]
+                subexpr = ExprBuilder(
+                    ArrayDiagonal,
+                    [
+                        ExprBuilder(
+                            ArrayTensorProduct,
+                            [
+                                ExprBuilder(_make_matrix, [l1]),
+                                hadam,
+                                ExprBuilder(_make_matrix, [l2]),
+                            ]
+                        ),
+                    *diagonal],
+
+                )
+                i._first_pointer_parent = subexpr.args[0].args[0].args
+                i._first_pointer_index = 0
+                i._second_pointer_parent = subexpr.args[0].args[2].args
+                i._second_pointer_index = 0
+                i._lines = [subexpr]
+                lines.append(i)
+
+        return lines
+
+
+# TODO Implement algorithm for rewriting Hadamard product as diagonal matrix
+# if matmul identy matrix is multiplied.
+def canonicalize(x):
+    """Canonicalize the Hadamard product ``x`` with mathematical properties.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, HadamardProduct
+    >>> from sympy import OneMatrix, ZeroMatrix
+    >>> from sympy.matrices.expressions.hadamard import canonicalize
+    >>> from sympy import init_printing
+    >>> init_printing(use_unicode=False)
+
+    >>> A = MatrixSymbol('A', 2, 2)
+    >>> B = MatrixSymbol('B', 2, 2)
+    >>> C = MatrixSymbol('C', 2, 2)
+
+    Hadamard product associativity:
+
+    >>> X = HadamardProduct(A, HadamardProduct(B, C))
+    >>> X
+    A.*(B.*C)
+    >>> canonicalize(X)
+    A.*B.*C
+
+    Hadamard product commutativity:
+
+    >>> X = HadamardProduct(A, B)
+    >>> Y = HadamardProduct(B, A)
+    >>> X
+    A.*B
+    >>> Y
+    B.*A
+    >>> canonicalize(X)
+    A.*B
+    >>> canonicalize(Y)
+    A.*B
+
+    Hadamard product identity:
+
+    >>> X = HadamardProduct(A, OneMatrix(2, 2))
+    >>> X
+    A.*1
+    >>> canonicalize(X)
+    A
+
+    Absorbing element of Hadamard product:
+
+    >>> X = HadamardProduct(A, ZeroMatrix(2, 2))
+    >>> X
+    A.*0
+    >>> canonicalize(X)
+    0
+
+    Rewriting to Hadamard Power
+
+    >>> X = HadamardProduct(A, A, A)
+    >>> X
+    A.*A.*A
+    >>> canonicalize(X)
+     .3
+    A
+
+    Notes
+    =====
+
+    As the Hadamard product is associative, nested products can be flattened.
+
+    The Hadamard product is commutative so that factors can be sorted for
+    canonical form.
+
+    A matrix of only ones is an identity for Hadamard product,
+    so every matrices of only ones can be removed.
+
+    Any zero matrix will make the whole product a zero matrix.
+
+    Duplicate elements can be collected and rewritten as HadamardPower
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
+    """
+    # Associativity
+    rule = condition(
+            lambda x: isinstance(x, HadamardProduct),
+            flatten
+        )
+    fun = exhaust(rule)
+    x = fun(x)
+
+    # Identity
+    fun = condition(
+            lambda x: isinstance(x, HadamardProduct),
+            rm_id(lambda x: isinstance(x, OneMatrix))
+        )
+    x = fun(x)
+
+    # Absorbing by Zero Matrix
+    def absorb(x):
+        if any(isinstance(c, ZeroMatrix) for c in x.args):
+            return ZeroMatrix(*x.shape)
+        else:
+            return x
+    fun = condition(
+            lambda x: isinstance(x, HadamardProduct),
+            absorb
+        )
+    x = fun(x)
+
+    # Rewriting with HadamardPower
+    if isinstance(x, HadamardProduct):
+        tally = Counter(x.args)
+
+        new_arg = []
+        for base, exp in tally.items():
+            if exp == 1:
+                new_arg.append(base)
+            else:
+                new_arg.append(HadamardPower(base, exp))
+
+        x = HadamardProduct(*new_arg)
+
+    # Commutativity
+    fun = condition(
+            lambda x: isinstance(x, HadamardProduct),
+            sort(default_sort_key)
+        )
+    x = fun(x)
+
+    # Unpacking
+    x = unpack(x)
+    return x
+
+
+def hadamard_power(base, exp):
+    base = sympify(base)
+    exp = sympify(exp)
+    if exp == 1:
+        return base
+    if not base.is_Matrix:
+        return base**exp
+    if exp.is_Matrix:
+        raise ValueError("cannot raise expression to a matrix")
+    return HadamardPower(base, exp)
+
+
+class HadamardPower(MatrixExpr):
+    r"""
+    Elementwise power of matrix expressions
+
+    Parameters
+    ==========
+
+    base : scalar or matrix
+
+    exp : scalar or matrix
+
+    Notes
+    =====
+
+    There are four definitions for the hadamard power which can be used.
+    Let's consider `A, B` as `(m, n)` matrices, and `a, b` as scalars.
+
+    Matrix raised to a scalar exponent:
+
+    .. math::
+        A^{\circ b} = \begin{bmatrix}
+        A_{0, 0}^b   & A_{0, 1}^b   & \cdots & A_{0, n-1}^b   \\
+        A_{1, 0}^b   & A_{1, 1}^b   & \cdots & A_{1, n-1}^b   \\
+        \vdots       & \vdots       & \ddots & \vdots         \\
+        A_{m-1, 0}^b & A_{m-1, 1}^b & \cdots & A_{m-1, n-1}^b
+        \end{bmatrix}
+
+    Scalar raised to a matrix exponent:
+
+    .. math::
+        a^{\circ B} = \begin{bmatrix}
+        a^{B_{0, 0}}   & a^{B_{0, 1}}   & \cdots & a^{B_{0, n-1}}   \\
+        a^{B_{1, 0}}   & a^{B_{1, 1}}   & \cdots & a^{B_{1, n-1}}   \\
+        \vdots         & \vdots         & \ddots & \vdots           \\
+        a^{B_{m-1, 0}} & a^{B_{m-1, 1}} & \cdots & a^{B_{m-1, n-1}}
+        \end{bmatrix}
+
+    Matrix raised to a matrix exponent:
+
+    .. math::
+        A^{\circ B} = \begin{bmatrix}
+        A_{0, 0}^{B_{0, 0}}     & A_{0, 1}^{B_{0, 1}}     &
+        \cdots & A_{0, n-1}^{B_{0, n-1}}     \\
+        A_{1, 0}^{B_{1, 0}}     & A_{1, 1}^{B_{1, 1}}     &
+        \cdots & A_{1, n-1}^{B_{1, n-1}}     \\
+        \vdots                  & \vdots                  &
+        \ddots & \vdots                      \\
+        A_{m-1, 0}^{B_{m-1, 0}} & A_{m-1, 1}^{B_{m-1, 1}} &
+        \cdots & A_{m-1, n-1}^{B_{m-1, n-1}}
+        \end{bmatrix}
+
+    Scalar raised to a scalar exponent:
+
+    .. math::
+        a^{\circ b} = a^b
+    """
+
+    def __new__(cls, base, exp):
+        base = sympify(base)
+        exp = sympify(exp)
+
+        if base.is_scalar and exp.is_scalar:
+            return base ** exp
+
+        if isinstance(base, MatrixExpr) and isinstance(exp, MatrixExpr):
+            validate(base, exp)
+
+        obj = super().__new__(cls, base, exp)
+        return obj
+
+    @property
+    def base(self):
+        return self._args[0]
+
+    @property
+    def exp(self):
+        return self._args[1]
+
+    @property
+    def shape(self):
+        if self.base.is_Matrix:
+            return self.base.shape
+        return self.exp.shape
+
+    def _entry(self, i, j, **kwargs):
+        base = self.base
+        exp = self.exp
+
+        if base.is_Matrix:
+            a = base._entry(i, j, **kwargs)
+        elif base.is_scalar:
+            a = base
+        else:
+            raise ValueError(
+                'The base {} must be a scalar or a matrix.'.format(base))
+
+        if exp.is_Matrix:
+            b = exp._entry(i, j, **kwargs)
+        elif exp.is_scalar:
+            b = exp
+        else:
+            raise ValueError(
+                'The exponent {} must be a scalar or a matrix.'.format(exp))
+
+        return a ** b
+
+    def _eval_transpose(self):
+        from sympy.matrices.expressions.transpose import transpose
+        return HadamardPower(transpose(self.base), self.exp)
+
+    def _eval_derivative(self, x):
+        dexp = self.exp.diff(x)
+        logbase = self.base.applyfunc(log)
+        dlbase = logbase.diff(x)
+        return hadamard_product(
+            dexp*logbase + self.exp*dlbase,
+            self
+        )
+
+    def _eval_derivative_matrix_lines(self, x):
+        from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
+        from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
+        from sympy.matrices.expressions.matexpr import _make_matrix
+
+        lr = self.base._eval_derivative_matrix_lines(x)
+        for i in lr:
+            diagonal = [(1, 2), (3, 4)]
+            diagonal = [e for j, e in enumerate(diagonal) if self.base.shape[j] != 1]
+            l1 = i._lines[i._first_line_index]
+            l2 = i._lines[i._second_line_index]
+            subexpr = ExprBuilder(
+                ArrayDiagonal,
+                [
+                    ExprBuilder(
+                        ArrayTensorProduct,
+                        [
+                            ExprBuilder(_make_matrix, [l1]),
+                            self.exp*hadamard_power(self.base, self.exp-1),
+                            ExprBuilder(_make_matrix, [l2]),
+                        ]
+                    ),
+                *diagonal],
+                validator=ArrayDiagonal._validate
+            )
+            i._first_pointer_parent = subexpr.args[0].args[0].args
+            i._first_pointer_index = 0
+            i._first_line_index = 0
+            i._second_pointer_parent = subexpr.args[0].args[2].args
+            i._second_pointer_index = 0
+            i._second_line_index = 0
+            i._lines = [subexpr]
+        return lr

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/inverse.py

@@ -0,0 +1,103 @@
+from sympy.core.sympify import _sympify
+from sympy.core import S, Basic
+
+from sympy.matrices.common import NonSquareMatrixError
+from sympy.matrices.expressions.matpow import MatPow
+
+
+class Inverse(MatPow):
+    """
+    The multiplicative inverse of a matrix expression
+
+    This is a symbolic object that simply stores its argument without
+    evaluating it. To actually compute the inverse, use the ``.inverse()``
+    method of matrices.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, Inverse
+    >>> A = MatrixSymbol('A', 3, 3)
+    >>> B = MatrixSymbol('B', 3, 3)
+    >>> Inverse(A)
+    A**(-1)
+    >>> A.inverse() == Inverse(A)
+    True
+    >>> (A*B).inverse()
+    B**(-1)*A**(-1)
+    >>> Inverse(A*B)
+    (A*B)**(-1)
+
+    """
+    is_Inverse = True
+    exp = S.NegativeOne
+
+    def __new__(cls, mat, exp=S.NegativeOne):
+        # exp is there to make it consistent with
+        # inverse.func(*inverse.args) == inverse
+        mat = _sympify(mat)
+        exp = _sympify(exp)
+        if not mat.is_Matrix:
+            raise TypeError("mat should be a matrix")
+        if mat.is_square is False:
+            raise NonSquareMatrixError("Inverse of non-square matrix %s" % mat)
+        return Basic.__new__(cls, mat, exp)
+
+    @property
+    def arg(self):
+        return self.args[0]
+
+    @property
+    def shape(self):
+        return self.arg.shape
+
+    def _eval_inverse(self):
+        return self.arg
+
+    def _eval_determinant(self):
+        from sympy.matrices.expressions.determinant import det
+        return 1/det(self.arg)
+
+    def doit(self, **hints):
+        if 'inv_expand' in hints and hints['inv_expand'] == False:
+            return self
+
+        arg = self.arg
+        if hints.get('deep', True):
+            arg = arg.doit(**hints)
+
+        return arg.inverse()
+
+    def _eval_derivative_matrix_lines(self, x):
+        arg = self.args[0]
+        lines = arg._eval_derivative_matrix_lines(x)
+        for line in lines:
+            line.first_pointer *= -self.T
+            line.second_pointer *= self
+        return lines
+
+
+from sympy.assumptions.ask import ask, Q
+from sympy.assumptions.refine import handlers_dict
+
+
+def refine_Inverse(expr, assumptions):
+    """
+    >>> from sympy import MatrixSymbol, Q, assuming, refine
+    >>> X = MatrixSymbol('X', 2, 2)
+    >>> X.I
+    X**(-1)
+    >>> with assuming(Q.orthogonal(X)):
+    ...     print(refine(X.I))
+    X.T
+    """
+    if ask(Q.orthogonal(expr), assumptions):
+        return expr.arg.T
+    elif ask(Q.unitary(expr), assumptions):
+        return expr.arg.conjugate()
+    elif ask(Q.singular(expr), assumptions):
+        raise ValueError("Inverse of singular matrix %s" % expr.arg)
+
+    return expr
+
+handlers_dict['Inverse'] = refine_Inverse

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/matadd.py

@@ -0,0 +1,155 @@
+from functools import reduce
+import operator
+
+from sympy.core import Basic, sympify
+from sympy.core.add import add, Add, _could_extract_minus_sign
+from sympy.core.sorting import default_sort_key
+from sympy.functions import adjoint
+from sympy.matrices.matrices import MatrixBase
+from sympy.matrices.expressions.transpose import transpose
+from sympy.strategies import (rm_id, unpack, flatten, sort, condition,
+    exhaust, do_one, glom)
+from sympy.matrices.expressions.matexpr import MatrixExpr
+from sympy.matrices.expressions.special import ZeroMatrix, GenericZeroMatrix
+from sympy.matrices.expressions._shape import validate_matadd_integer as validate
+from sympy.utilities.iterables import sift
+from sympy.utilities.exceptions import sympy_deprecation_warning
+
+# XXX: MatAdd should perhaps not subclass directly from Add
+class MatAdd(MatrixExpr, Add):
+    """A Sum of Matrix Expressions
+
+    MatAdd inherits from and operates like SymPy Add
+
+    Examples
+    ========
+
+    >>> from sympy import MatAdd, MatrixSymbol
+    >>> A = MatrixSymbol('A', 5, 5)
+    >>> B = MatrixSymbol('B', 5, 5)
+    >>> C = MatrixSymbol('C', 5, 5)
+    >>> MatAdd(A, B, C)
+    A + B + C
+    """
+    is_MatAdd = True
+
+    identity = GenericZeroMatrix()
+
+    def __new__(cls, *args, evaluate=False, check=None, _sympify=True):
+        if not args:
+            return cls.identity
+
+        # This must be removed aggressively in the constructor to avoid
+        # TypeErrors from GenericZeroMatrix().shape
+        args = list(filter(lambda i: cls.identity != i, args))
+        if _sympify:
+            args = list(map(sympify, args))
+
+        if not all(isinstance(arg, MatrixExpr) for arg in args):
+            raise TypeError("Mix of Matrix and Scalar symbols")
+
+        obj = Basic.__new__(cls, *args)
+
+        if check is not None:
+            sympy_deprecation_warning(
+                "Passing check to MatAdd is deprecated and the check argument will be removed in a future version.",
+                deprecated_since_version="1.11",
+                active_deprecations_target='remove-check-argument-from-matrix-operations')
+
+        if check is not False:
+            validate(*args)
+
+        if evaluate:
+            obj = cls._evaluate(obj)
+
+        return obj
+
+    @classmethod
+    def _evaluate(cls, expr):
+        return canonicalize(expr)
+
+    @property
+    def shape(self):
+        return self.args[0].shape
+
+    def could_extract_minus_sign(self):
+        return _could_extract_minus_sign(self)
+
+    def expand(self, **kwargs):
+        expanded = super(MatAdd, self).expand(**kwargs)
+        return self._evaluate(expanded)
+
+    def _entry(self, i, j, **kwargs):
+        return Add(*[arg._entry(i, j, **kwargs) for arg in self.args])
+
+    def _eval_transpose(self):
+        return MatAdd(*[transpose(arg) for arg in self.args]).doit()
+
+    def _eval_adjoint(self):
+        return MatAdd(*[adjoint(arg) for arg in self.args]).doit()
+
+    def _eval_trace(self):
+        from .trace import trace
+        return Add(*[trace(arg) for arg in self.args]).doit()
+
+    def doit(self, **hints):
+        deep = hints.get('deep', True)
+        if deep:
+            args = [arg.doit(**hints) for arg in self.args]
+        else:
+            args = self.args
+        return canonicalize(MatAdd(*args))
+
+    def _eval_derivative_matrix_lines(self, x):
+        add_lines = [arg._eval_derivative_matrix_lines(x) for arg in self.args]
+        return [j for i in add_lines for j in i]
+
+add.register_handlerclass((Add, MatAdd), MatAdd)
+
+
+factor_of = lambda arg: arg.as_coeff_mmul()[0]
+matrix_of = lambda arg: unpack(arg.as_coeff_mmul()[1])
+def combine(cnt, mat):
+    if cnt == 1:
+        return mat
+    else:
+        return cnt * mat
+
+
+def merge_explicit(matadd):
+    """ Merge explicit MatrixBase arguments
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, eye, Matrix, MatAdd, pprint
+    >>> from sympy.matrices.expressions.matadd import merge_explicit
+    >>> A = MatrixSymbol('A', 2, 2)
+    >>> B = eye(2)
+    >>> C = Matrix([[1, 2], [3, 4]])
+    >>> X = MatAdd(A, B, C)
+    >>> pprint(X)
+        [1  0]   [1  2]
+    A + [    ] + [    ]
+        [0  1]   [3  4]
+    >>> pprint(merge_explicit(X))
+        [2  2]
+    A + [    ]
+        [3  5]
+    """
+    groups = sift(matadd.args, lambda arg: isinstance(arg, MatrixBase))
+    if len(groups[True]) > 1:
+        return MatAdd(*(groups[False] + [reduce(operator.add, groups[True])]))
+    else:
+        return matadd
+
+
+rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)),
+         unpack,
+         flatten,
+         glom(matrix_of, factor_of, combine),
+         merge_explicit,
+         sort(default_sort_key))
+
+canonicalize = exhaust(condition(lambda x: isinstance(x, MatAdd),
+                                 do_one(*rules)))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/matexpr.py

@@ -0,0 +1,885 @@
+from __future__ import annotations
+from functools import wraps
+
+from sympy.core import S, Integer, Basic, Mul, Add
+from sympy.core.assumptions import check_assumptions
+from sympy.core.decorators import call_highest_priority
+from sympy.core.expr import Expr, ExprBuilder
+from sympy.core.logic import FuzzyBool
+from sympy.core.symbol import Str, Dummy, symbols, Symbol
+from sympy.core.sympify import SympifyError, _sympify
+from sympy.external.gmpy import SYMPY_INTS
+from sympy.functions import conjugate, adjoint
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.matrices.common import NonSquareMatrixError
+from sympy.matrices.matrices import MatrixKind, MatrixBase
+from sympy.multipledispatch import dispatch
+from sympy.utilities.misc import filldedent
+
+
+def _sympifyit(arg, retval=None):
+    # This version of _sympifyit sympifies MutableMatrix objects
+    def deco(func):
+        @wraps(func)
+        def __sympifyit_wrapper(a, b):
+            try:
+                b = _sympify(b)
+                return func(a, b)
+            except SympifyError:
+                return retval
+
+        return __sympifyit_wrapper
+
+    return deco
+
+
+class MatrixExpr(Expr):
+    """Superclass for Matrix Expressions
+
+    MatrixExprs represent abstract matrices, linear transformations represented
+    within a particular basis.
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol
+    >>> A = MatrixSymbol('A', 3, 3)
+    >>> y = MatrixSymbol('y', 3, 1)
+    >>> x = (A.T*A).I * A * y
+
+    See Also
+    ========
+
+    MatrixSymbol, MatAdd, MatMul, Transpose, Inverse
+    """
+    __slots__: tuple[str, ...] = ()
+
+    # Should not be considered iterable by the
+    # sympy.utilities.iterables.iterable function. Subclass that actually are
+    # iterable (i.e., explicit matrices) should set this to True.
+    _iterable = False
+
+    _op_priority = 11.0
+
+    is_Matrix: bool = True
+    is_MatrixExpr: bool = True
+    is_Identity: FuzzyBool = None
+    is_Inverse = False
+    is_Transpose = False
+    is_ZeroMatrix = False
+    is_MatAdd = False
+    is_MatMul = False
+
+    is_commutative = False
+    is_number = False
+    is_symbol = False
+    is_scalar = False
+
+    kind: MatrixKind = MatrixKind()
+
+    def __new__(cls, *args, **kwargs):
+        args = map(_sympify, args)
+        return Basic.__new__(cls, *args, **kwargs)
+
+    # The following is adapted from the core Expr object
+
+    @property
+    def shape(self) -> tuple[Expr, Expr]:
+        raise NotImplementedError
+
+    @property
+    def _add_handler(self):
+        return MatAdd
+
+    @property
+    def _mul_handler(self):
+        return MatMul
+
+    def __neg__(self):
+        return MatMul(S.NegativeOne, self).doit()
+
+    def __abs__(self):
+        raise NotImplementedError
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__radd__')
+    def __add__(self, other):
+        return MatAdd(self, other).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__add__')
+    def __radd__(self, other):
+        return MatAdd(other, self).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rsub__')
+    def __sub__(self, other):
+        return MatAdd(self, -other).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__sub__')
+    def __rsub__(self, other):
+        return MatAdd(other, -self).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rmul__')
+    def __mul__(self, other):
+        return MatMul(self, other).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rmul__')
+    def __matmul__(self, other):
+        return MatMul(self, other).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__mul__')
+    def __rmul__(self, other):
+        return MatMul(other, self).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__mul__')
+    def __rmatmul__(self, other):
+        return MatMul(other, self).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rpow__')
+    def __pow__(self, other):
+        return MatPow(self, other).doit()
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__pow__')
+    def __rpow__(self, other):
+        raise NotImplementedError("Matrix Power not defined")
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__rtruediv__')
+    def __truediv__(self, other):
+        return self * other**S.NegativeOne
+
+    @_sympifyit('other', NotImplemented)
+    @call_highest_priority('__truediv__')
+    def __rtruediv__(self, other):
+        raise NotImplementedError()
+        #return MatMul(other, Pow(self, S.NegativeOne))
+
+    @property
+    def rows(self):
+        return self.shape[0]
+
+    @property
+    def cols(self):
+        return self.shape[1]
+
+    @property
+    def is_square(self) -> bool | None:
+        rows, cols = self.shape
+        if isinstance(rows, Integer) and isinstance(cols, Integer):
+            return rows == cols
+        if rows == cols:
+            return True
+        return None
+
+    def _eval_conjugate(self):
+        from sympy.matrices.expressions.adjoint import Adjoint
+        return Adjoint(Transpose(self))
+
+    def as_real_imag(self, deep=True, **hints):
+        return self._eval_as_real_imag()
+
+    def _eval_as_real_imag(self):
+        real = S.Half * (self + self._eval_conjugate())
+        im = (self - self._eval_conjugate())/(2*S.ImaginaryUnit)
+        return (real, im)
+
+    def _eval_inverse(self):
+        return Inverse(self)
+
+    def _eval_determinant(self):
+        return Determinant(self)
+
+    def _eval_transpose(self):
+        return Transpose(self)
+
+    def _eval_power(self, exp):
+        """
+        Override this in sub-classes to implement simplification of powers.  The cases where the exponent
+        is -1, 0, 1 are already covered in MatPow.doit(), so implementations can exclude these cases.
+        """
+        return MatPow(self, exp)
+
+    def _eval_simplify(self, **kwargs):
+        if self.is_Atom:
+            return self
+        else:
+            from sympy.simplify import simplify
+            return self.func(*[simplify(x, **kwargs) for x in self.args])
+
+    def _eval_adjoint(self):
+        from sympy.matrices.expressions.adjoint import Adjoint
+        return Adjoint(self)
+
+    def _eval_derivative_n_times(self, x, n):
+        return Basic._eval_derivative_n_times(self, x, n)
+
+    def _eval_derivative(self, x):
+        # `x` is a scalar:
+        if self.has(x):
+            # See if there are other methods using it:
+            return super()._eval_derivative(x)
+        else:
+            return ZeroMatrix(*self.shape)
+
+    @classmethod
+    def _check_dim(cls, dim):
+        """Helper function to check invalid matrix dimensions"""
+        ok = check_assumptions(dim, integer=True, nonnegative=True)
+        if ok is False:
+            raise ValueError(
+                "The dimension specification {} should be "
+                "a nonnegative integer.".format(dim))
+
+
+    def _entry(self, i, j, **kwargs):
+        raise NotImplementedError(
+            "Indexing not implemented for %s" % self.__class__.__name__)
+
+    def adjoint(self):
+        return adjoint(self)
+
+    def as_coeff_Mul(self, rational=False):
+        """Efficiently extract the coefficient of a product."""
+        return S.One, self
+
+    def conjugate(self):
+        return conjugate(self)
+
+    def transpose(self):
+        from sympy.matrices.expressions.transpose import transpose
+        return transpose(self)
+
+    @property
+    def T(self):
+        '''Matrix transposition'''
+        return self.transpose()
+
+    def inverse(self):
+        if self.is_square is False:
+            raise NonSquareMatrixError('Inverse of non-square matrix')
+        return self._eval_inverse()
+
+    def inv(self):
+        return self.inverse()
+
+    def det(self):
+        from sympy.matrices.expressions.determinant import det
+        return det(self)
+
+    @property
+    def I(self):
+        return self.inverse()
+
+    def valid_index(self, i, j):
+        def is_valid(idx):
+            return isinstance(idx, (int, Integer, Symbol, Expr))
+        return (is_valid(i) and is_valid(j) and
+                (self.rows is None or
+                (i >= -self.rows) != False and (i < self.rows) != False) and
+                (j >= -self.cols) != False and (j < self.cols) != False)
+
+    def __getitem__(self, key):
+        if not isinstance(key, tuple) and isinstance(key, slice):
+            from sympy.matrices.expressions.slice import MatrixSlice
+            return MatrixSlice(self, key, (0, None, 1))
+        if isinstance(key, tuple) and len(key) == 2:
+            i, j = key
+            if isinstance(i, slice) or isinstance(j, slice):
+                from sympy.matrices.expressions.slice import MatrixSlice
+                return MatrixSlice(self, i, j)
+            i, j = _sympify(i), _sympify(j)
+            if self.valid_index(i, j) != False:
+                return self._entry(i, j)
+            else:
+                raise IndexError("Invalid indices (%s, %s)" % (i, j))
+        elif isinstance(key, (SYMPY_INTS, Integer)):
+            # row-wise decomposition of matrix
+            rows, cols = self.shape
+            # allow single indexing if number of columns is known
+            if not isinstance(cols, Integer):
+                raise IndexError(filldedent('''
+                    Single indexing is only supported when the number
+                    of columns is known.'''))
+            key = _sympify(key)
+            i = key // cols
+            j = key % cols
+            if self.valid_index(i, j) != False:
+                return self._entry(i, j)
+            else:
+                raise IndexError("Invalid index %s" % key)
+        elif isinstance(key, (Symbol, Expr)):
+            raise IndexError(filldedent('''
+                Only integers may be used when addressing the matrix
+                with a single index.'''))
+        raise IndexError("Invalid index, wanted %s[i,j]" % self)
+
+    def _is_shape_symbolic(self) -> bool:
+        return (not isinstance(self.rows, (SYMPY_INTS, Integer))
+            or not isinstance(self.cols, (SYMPY_INTS, Integer)))
+
+    def as_explicit(self):
+        """
+        Returns a dense Matrix with elements represented explicitly
+
+        Returns an object of type ImmutableDenseMatrix.
+
+        Examples
+        ========
+
+        >>> from sympy import Identity
+        >>> I = Identity(3)
+        >>> I
+        I
+        >>> I.as_explicit()
+        Matrix([
+        [1, 0, 0],
+        [0, 1, 0],
+        [0, 0, 1]])
+
+        See Also
+        ========
+        as_mutable: returns mutable Matrix type
+
+        """
+        if self._is_shape_symbolic():
+            raise ValueError(
+                'Matrix with symbolic shape '
+                'cannot be represented explicitly.')
+        from sympy.matrices.immutable import ImmutableDenseMatrix
+        return ImmutableDenseMatrix([[self[i, j]
+                            for j in range(self.cols)]
+                            for i in range(self.rows)])
+
+    def as_mutable(self):
+        """
+        Returns a dense, mutable matrix with elements represented explicitly
+
+        Examples
+        ========
+
+        >>> from sympy import Identity
+        >>> I = Identity(3)
+        >>> I
+        I
+        >>> I.shape
+        (3, 3)
+        >>> I.as_mutable()
+        Matrix([
+        [1, 0, 0],
+        [0, 1, 0],
+        [0, 0, 1]])
+
+        See Also
+        ========
+        as_explicit: returns ImmutableDenseMatrix
+        """
+        return self.as_explicit().as_mutable()
+
+    def __array__(self):
+        from numpy import empty
+        a = empty(self.shape, dtype=object)
+        for i in range(self.rows):
+            for j in range(self.cols):
+                a[i, j] = self[i, j]
+        return a
+
+    def equals(self, other):
+        """
+        Test elementwise equality between matrices, potentially of different
+        types
+
+        >>> from sympy import Identity, eye
+        >>> Identity(3).equals(eye(3))
+        True
+        """
+        return self.as_explicit().equals(other)
+
+    def canonicalize(self):
+        return self
+
+    def as_coeff_mmul(self):
+        return S.One, MatMul(self)
+
+    @staticmethod
+    def from_index_summation(expr, first_index=None, last_index=None, dimensions=None):
+        r"""
+        Parse expression of matrices with explicitly summed indices into a
+        matrix expression without indices, if possible.
+
+        This transformation expressed in mathematical notation:
+
+        `\sum_{j=0}^{N-1} A_{i,j} B_{j,k} \Longrightarrow \mathbf{A}\cdot \mathbf{B}`
+
+        Optional parameter ``first_index``: specify which free index to use as
+        the index starting the expression.
+
+        Examples
+        ========
+
+        >>> from sympy import MatrixSymbol, MatrixExpr, Sum
+        >>> from sympy.abc import i, j, k, l, N
+        >>> A = MatrixSymbol("A", N, N)
+        >>> B = MatrixSymbol("B", N, N)
+        >>> expr = Sum(A[i, j]*B[j, k], (j, 0, N-1))
+        >>> MatrixExpr.from_index_summation(expr)
+        A*B
+
+        Transposition is detected:
+
+        >>> expr = Sum(A[j, i]*B[j, k], (j, 0, N-1))
+        >>> MatrixExpr.from_index_summation(expr)
+        A.T*B
+
+        Detect the trace:
+
+        >>> expr = Sum(A[i, i], (i, 0, N-1))
+        >>> MatrixExpr.from_index_summation(expr)
+        Trace(A)
+
+        More complicated expressions:
+
+        >>> expr = Sum(A[i, j]*B[k, j]*A[l, k], (j, 0, N-1), (k, 0, N-1))
+        >>> MatrixExpr.from_index_summation(expr)
+        A*B.T*A.T
+        """
+        from sympy.tensor.array.expressions.from_indexed_to_array import convert_indexed_to_array
+        from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix
+        first_indices = []
+        if first_index is not None:
+            first_indices.append(first_index)
+        if last_index is not None:
+            first_indices.append(last_index)
+        arr = convert_indexed_to_array(expr, first_indices=first_indices)
+        return convert_array_to_matrix(arr)
+
+    def applyfunc(self, func):
+        from .applyfunc import ElementwiseApplyFunction
+        return ElementwiseApplyFunction(func, self)
+
+
+@dispatch(MatrixExpr, Expr)
+def _eval_is_eq(lhs, rhs): # noqa:F811
+    return False
+
+@dispatch(MatrixExpr, MatrixExpr)  # type: ignore
+def _eval_is_eq(lhs, rhs): # noqa:F811
+    if lhs.shape != rhs.shape:
+        return False
+    if (lhs - rhs).is_ZeroMatrix:
+        return True
+
+def get_postprocessor(cls):
+    def _postprocessor(expr):
+        # To avoid circular imports, we can't have MatMul/MatAdd on the top level
+        mat_class = {Mul: MatMul, Add: MatAdd}[cls]
+        nonmatrices = []
+        matrices = []
+        for term in expr.args:
+            if isinstance(term, MatrixExpr):
+                matrices.append(term)
+            else:
+                nonmatrices.append(term)
+
+        if not matrices:
+            return cls._from_args(nonmatrices)
+
+        if nonmatrices:
+            if cls == Mul:
+                for i in range(len(matrices)):
+                    if not matrices[i].is_MatrixExpr:
+                        # If one of the matrices explicit, absorb the scalar into it
+                        # (doit will combine all explicit matrices into one, so it
+                        # doesn't matter which)
+                        matrices[i] = matrices[i].__mul__(cls._from_args(nonmatrices))
+                        nonmatrices = []
+                        break
+
+            else:
+                # Maintain the ability to create Add(scalar, matrix) without
+                # raising an exception. That way different algorithms can
+                # replace matrix expressions with non-commutative symbols to
+                # manipulate them like non-commutative scalars.
+                return cls._from_args(nonmatrices + [mat_class(*matrices).doit(deep=False)])
+
+        if mat_class == MatAdd:
+            return mat_class(*matrices).doit(deep=False)
+        return mat_class(cls._from_args(nonmatrices), *matrices).doit(deep=False)
+    return _postprocessor
+
+
+Basic._constructor_postprocessor_mapping[MatrixExpr] = {
+    "Mul": [get_postprocessor(Mul)],
+    "Add": [get_postprocessor(Add)],
+}
+
+
+def _matrix_derivative(expr, x, old_algorithm=False):
+
+    if isinstance(expr, MatrixBase) or isinstance(x, MatrixBase):
+        # Do not use array expressions for explicit matrices:
+        old_algorithm = True
+
+    if old_algorithm:
+        return _matrix_derivative_old_algorithm(expr, x)
+
+    from sympy.tensor.array.expressions.from_matrix_to_array import convert_matrix_to_array
+    from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive
+    from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix
+
+    array_expr = convert_matrix_to_array(expr)
+    diff_array_expr = array_derive(array_expr, x)
+    diff_matrix_expr = convert_array_to_matrix(diff_array_expr)
+    return diff_matrix_expr
+
+
+def _matrix_derivative_old_algorithm(expr, x):
+    from sympy.tensor.array.array_derivatives import ArrayDerivative
+    lines = expr._eval_derivative_matrix_lines(x)
+
+    parts = [i.build() for i in lines]
+
+    from sympy.tensor.array.expressions.from_array_to_matrix import convert_array_to_matrix
+
+    parts = [[convert_array_to_matrix(j) for j in i] for i in parts]
+
+    def _get_shape(elem):
+        if isinstance(elem, MatrixExpr):
+            return elem.shape
+        return 1, 1
+
+    def get_rank(parts):
+        return sum([j not in (1, None) for i in parts for j in _get_shape(i)])
+
+    ranks = [get_rank(i) for i in parts]
+    rank = ranks[0]
+
+    def contract_one_dims(parts):
+        if len(parts) == 1:
+            return parts[0]
+        else:
+            p1, p2 = parts[:2]
+            if p2.is_Matrix:
+                p2 = p2.T
+            if p1 == Identity(1):
+                pbase = p2
+            elif p2 == Identity(1):
+                pbase = p1
+            else:
+                pbase = p1*p2
+            if len(parts) == 2:
+                return pbase
+            else:  # len(parts) > 2
+                if pbase.is_Matrix:
+                    raise ValueError("")
+                return pbase*Mul.fromiter(parts[2:])
+
+    if rank <= 2:
+        return Add.fromiter([contract_one_dims(i) for i in parts])
+
+    return ArrayDerivative(expr, x)
+
+
+class MatrixElement(Expr):
+    parent = property(lambda self: self.args[0])
+    i = property(lambda self: self.args[1])
+    j = property(lambda self: self.args[2])
+    _diff_wrt = True
+    is_symbol = True
+    is_commutative = True
+
+    def __new__(cls, name, n, m):
+        n, m = map(_sympify, (n, m))
+        from sympy.matrices.matrices import MatrixBase
+        if isinstance(name, str):
+            name = Symbol(name)
+        else:
+            if isinstance(name, MatrixBase):
+                if n.is_Integer and m.is_Integer:
+                    return name[n, m]
+                name = _sympify(name)  # change mutable into immutable
+            else:
+                name = _sympify(name)
+                if not isinstance(name.kind, MatrixKind):
+                    raise TypeError("First argument of MatrixElement should be a matrix")
+            if not getattr(name, 'valid_index', lambda n, m: True)(n, m):
+                raise IndexError('indices out of range')
+        obj = Expr.__new__(cls, name, n, m)
+        return obj
+
+    @property
+    def symbol(self):
+        return self.args[0]
+
+    def doit(self, **hints):
+        deep = hints.get('deep', True)
+        if deep:
+            args = [arg.doit(**hints) for arg in self.args]
+        else:
+            args = self.args
+        return args[0][args[1], args[2]]
+
+    @property
+    def indices(self):
+        return self.args[1:]
+
+    def _eval_derivative(self, v):
+
+        if not isinstance(v, MatrixElement):
+            from sympy.matrices.matrices import MatrixBase
+            if isinstance(self.parent, MatrixBase):
+                return self.parent.diff(v)[self.i, self.j]
+            return S.Zero
+
+        M = self.args[0]
+
+        m, n = self.parent.shape
+
+        if M == v.args[0]:
+            return KroneckerDelta(self.args[1], v.args[1], (0, m-1)) * \
+                   KroneckerDelta(self.args[2], v.args[2], (0, n-1))
+
+        if isinstance(M, Inverse):
+            from sympy.concrete.summations import Sum
+            i, j = self.args[1:]
+            i1, i2 = symbols("z1, z2", cls=Dummy)
+            Y = M.args[0]
+            r1, r2 = Y.shape
+            return -Sum(M[i, i1]*Y[i1, i2].diff(v)*M[i2, j], (i1, 0, r1-1), (i2, 0, r2-1))
+
+        if self.has(v.args[0]):
+            return None
+
+        return S.Zero
+
+
+class MatrixSymbol(MatrixExpr):
+    """Symbolic representation of a Matrix object
+
+    Creates a SymPy Symbol to represent a Matrix. This matrix has a shape and
+    can be included in Matrix Expressions
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, Identity
+    >>> A = MatrixSymbol('A', 3, 4) # A 3 by 4 Matrix
+    >>> B = MatrixSymbol('B', 4, 3) # A 4 by 3 Matrix
+    >>> A.shape
+    (3, 4)
+    >>> 2*A*B + Identity(3)
+    I + 2*A*B
+    """
+    is_commutative = False
+    is_symbol = True
+    _diff_wrt = True
+
+    def __new__(cls, name, n, m):
+        n, m = _sympify(n), _sympify(m)
+
+        cls._check_dim(m)
+        cls._check_dim(n)
+
+        if isinstance(name, str):
+            name = Str(name)
+        obj = Basic.__new__(cls, name, n, m)
+        return obj
+
+    @property
+    def shape(self):
+        return self.args[1], self.args[2]
+
+    @property
+    def name(self):
+        return self.args[0].name
+
+    def _entry(self, i, j, **kwargs):
+        return MatrixElement(self, i, j)
+
+    @property
+    def free_symbols(self):
+        return {self}
+
+    def _eval_simplify(self, **kwargs):
+        return self
+
+    def _eval_derivative(self, x):
+        # x is a scalar:
+        return ZeroMatrix(self.shape[0], self.shape[1])
+
+    def _eval_derivative_matrix_lines(self, x):
+        if self != x:
+            first = ZeroMatrix(x.shape[0], self.shape[0]) if self.shape[0] != 1 else S.Zero
+            second = ZeroMatrix(x.shape[1], self.shape[1]) if self.shape[1] != 1 else S.Zero
+            return [_LeftRightArgs(
+                [first, second],
+            )]
+        else:
+            first = Identity(self.shape[0]) if self.shape[0] != 1 else S.One
+            second = Identity(self.shape[1]) if self.shape[1] != 1 else S.One
+            return [_LeftRightArgs(
+                [first, second],
+            )]
+
+
+def matrix_symbols(expr):
+    return [sym for sym in expr.free_symbols if sym.is_Matrix]
+
+
+class _LeftRightArgs:
+    r"""
+    Helper class to compute matrix derivatives.
+
+    The logic: when an expression is derived by a matrix `X_{mn}`, two lines of
+    matrix multiplications are created: the one contracted to `m` (first line),
+    and the one contracted to `n` (second line).
+
+    Transposition flips the side by which new matrices are connected to the
+    lines.
+
+    The trace connects the end of the two lines.
+    """
+
+    def __init__(self, lines, higher=S.One):
+        self._lines = list(lines)
+        self._first_pointer_parent = self._lines
+        self._first_pointer_index = 0
+        self._first_line_index = 0
+        self._second_pointer_parent = self._lines
+        self._second_pointer_index = 1
+        self._second_line_index = 1
+        self.higher = higher
+
+    @property
+    def first_pointer(self):
+       return self._first_pointer_parent[self._first_pointer_index]
+
+    @first_pointer.setter
+    def first_pointer(self, value):
+        self._first_pointer_parent[self._first_pointer_index] = value
+
+    @property
+    def second_pointer(self):
+        return self._second_pointer_parent[self._second_pointer_index]
+
+    @second_pointer.setter
+    def second_pointer(self, value):
+        self._second_pointer_parent[self._second_pointer_index] = value
+
+    def __repr__(self):
+        built = [self._build(i) for i in self._lines]
+        return "_LeftRightArgs(lines=%s, higher=%s)" % (
+            built,
+            self.higher,
+        )
+
+    def transpose(self):
+        self._first_pointer_parent, self._second_pointer_parent = self._second_pointer_parent, self._first_pointer_parent
+        self._first_pointer_index, self._second_pointer_index = self._second_pointer_index, self._first_pointer_index
+        self._first_line_index, self._second_line_index = self._second_line_index, self._first_line_index
+        return self
+
+    @staticmethod
+    def _build(expr):
+        if isinstance(expr, ExprBuilder):
+            return expr.build()
+        if isinstance(expr, list):
+            if len(expr) == 1:
+                return expr[0]
+            else:
+                return expr[0](*[_LeftRightArgs._build(i) for i in expr[1]])
+        else:
+            return expr
+
+    def build(self):
+        data = [self._build(i) for i in self._lines]
+        if self.higher != 1:
+            data += [self._build(self.higher)]
+        data = list(data)
+        return data
+
+    def matrix_form(self):
+        if self.first != 1 and self.higher != 1:
+            raise ValueError("higher dimensional array cannot be represented")
+
+        def _get_shape(elem):
+            if isinstance(elem, MatrixExpr):
+                return elem.shape
+            return (None, None)
+
+        if _get_shape(self.first)[1] != _get_shape(self.second)[1]:
+            # Remove one-dimensional identity matrices:
+            # (this is needed by `a.diff(a)` where `a` is a vector)
+            if _get_shape(self.second) == (1, 1):
+                return self.first*self.second[0, 0]
+            if _get_shape(self.first) == (1, 1):
+                return self.first[1, 1]*self.second.T
+            raise ValueError("incompatible shapes")
+        if self.first != 1:
+            return self.first*self.second.T
+        else:
+            return self.higher
+
+    def rank(self):
+        """
+        Number of dimensions different from trivial (warning: not related to
+        matrix rank).
+        """
+        rank = 0
+        if self.first != 1:
+            rank += sum([i != 1 for i in self.first.shape])
+        if self.second != 1:
+            rank += sum([i != 1 for i in self.second.shape])
+        if self.higher != 1:
+            rank += 2
+        return rank
+
+    def _multiply_pointer(self, pointer, other):
+        from ...tensor.array.expressions.array_expressions import ArrayTensorProduct
+        from ...tensor.array.expressions.array_expressions import ArrayContraction
+
+        subexpr = ExprBuilder(
+            ArrayContraction,
+            [
+                ExprBuilder(
+                    ArrayTensorProduct,
+                    [
+                        pointer,
+                        other
+                    ]
+                ),
+                (1, 2)
+            ],
+            validator=ArrayContraction._validate
+        )
+
+        return subexpr
+
+    def append_first(self, other):
+        self.first_pointer *= other
+
+    def append_second(self, other):
+        self.second_pointer *= other
+
+
+def _make_matrix(x):
+    from sympy.matrices.immutable import ImmutableDenseMatrix
+    if isinstance(x, MatrixExpr):
+        return x
+    return ImmutableDenseMatrix([[x]])
+
+
+from .matmul import MatMul
+from .matadd import MatAdd
+from .matpow import MatPow
+from .transpose import Transpose
+from .inverse import Inverse
+from .special import ZeroMatrix, Identity
+from .determinant import Determinant

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/matmul.py

@@ -0,0 +1,498 @@
+from sympy.assumptions.ask import ask, Q
+from sympy.assumptions.refine import handlers_dict
+from sympy.core import Basic, sympify, S
+from sympy.core.mul import mul, Mul
+from sympy.core.numbers import Number, Integer
+from sympy.core.symbol import Dummy
+from sympy.functions import adjoint
+from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
+        do_one, new)
+from sympy.matrices.common import NonInvertibleMatrixError
+from sympy.matrices.matrices import MatrixBase
+from sympy.utilities.exceptions import sympy_deprecation_warning
+from sympy.matrices.expressions._shape import validate_matmul_integer as validate
+
+from .inverse import Inverse
+from .matexpr import MatrixExpr
+from .matpow import MatPow
+from .transpose import transpose
+from .permutation import PermutationMatrix
+from .special import ZeroMatrix, Identity, GenericIdentity, OneMatrix
+
+
+# XXX: MatMul should perhaps not subclass directly from Mul
+class MatMul(MatrixExpr, Mul):
+    """
+    A product of matrix expressions
+
+    Examples
+    ========
+
+    >>> from sympy import MatMul, MatrixSymbol
+    >>> A = MatrixSymbol('A', 5, 4)
+    >>> B = MatrixSymbol('B', 4, 3)
+    >>> C = MatrixSymbol('C', 3, 6)
+    >>> MatMul(A, B, C)
+    A*B*C
+    """
+    is_MatMul = True
+
+    identity = GenericIdentity()
+
+    def __new__(cls, *args, evaluate=False, check=None, _sympify=True):
+        if not args:
+            return cls.identity
+
+        # This must be removed aggressively in the constructor to avoid
+        # TypeErrors from GenericIdentity().shape
+        args = list(filter(lambda i: cls.identity != i, args))
+        if _sympify:
+            args = list(map(sympify, args))
+        obj = Basic.__new__(cls, *args)
+        factor, matrices = obj.as_coeff_matrices()
+
+        if check is not None:
+            sympy_deprecation_warning(
+                "Passing check to MatMul is deprecated and the check argument will be removed in a future version.",
+                deprecated_since_version="1.11",
+                active_deprecations_target='remove-check-argument-from-matrix-operations')
+
+        if check is not False:
+            validate(*matrices)
+
+        if not matrices:
+            # Should it be
+            #
+            # return Basic.__neq__(cls, factor, GenericIdentity()) ?
+            return factor
+
+        if evaluate:
+            return cls._evaluate(obj)
+
+        return obj
+
+    @classmethod
+    def _evaluate(cls, expr):
+        return canonicalize(expr)
+
+    @property
+    def shape(self):
+        matrices = [arg for arg in self.args if arg.is_Matrix]
+        return (matrices[0].rows, matrices[-1].cols)
+
+    def _entry(self, i, j, expand=True, **kwargs):
+        # Avoid cyclic imports
+        from sympy.concrete.summations import Sum
+        from sympy.matrices.immutable import ImmutableMatrix
+
+        coeff, matrices = self.as_coeff_matrices()
+
+        if len(matrices) == 1:  # situation like 2*X, matmul is just X
+            return coeff * matrices[0][i, j]
+
+        indices = [None]*(len(matrices) + 1)
+        ind_ranges = [None]*(len(matrices) - 1)
+        indices[0] = i
+        indices[-1] = j
+
+        def f():
+            counter = 1
+            while True:
+                yield Dummy("i_%i" % counter)
+                counter += 1
+
+        dummy_generator = kwargs.get("dummy_generator", f())
+
+        for i in range(1, len(matrices)):
+            indices[i] = next(dummy_generator)
+
+        for i, arg in enumerate(matrices[:-1]):
+            ind_ranges[i] = arg.shape[1] - 1
+        matrices = [arg._entry(indices[i], indices[i+1], dummy_generator=dummy_generator) for i, arg in enumerate(matrices)]
+        expr_in_sum = Mul.fromiter(matrices)
+        if any(v.has(ImmutableMatrix) for v in matrices):
+            expand = True
+        result = coeff*Sum(
+                expr_in_sum,
+                *zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
+            )
+
+        # Don't waste time in result.doit() if the sum bounds are symbolic
+        if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
+            expand = False
+        return result.doit() if expand else result
+
+    def as_coeff_matrices(self):
+        scalars = [x for x in self.args if not x.is_Matrix]
+        matrices = [x for x in self.args if x.is_Matrix]
+        coeff = Mul(*scalars)
+        if coeff.is_commutative is False:
+            raise NotImplementedError("noncommutative scalars in MatMul are not supported.")
+
+        return coeff, matrices
+
+    def as_coeff_mmul(self):
+        coeff, matrices = self.as_coeff_matrices()
+        return coeff, MatMul(*matrices)
+
+    def expand(self, **kwargs):
+        expanded = super(MatMul, self).expand(**kwargs)
+        return self._evaluate(expanded)
+
+    def _eval_transpose(self):
+        """Transposition of matrix multiplication.
+
+        Notes
+        =====
+
+        The following rules are applied.
+
+        Transposition for matrix multiplied with another matrix:
+        `\\left(A B\\right)^{T} = B^{T} A^{T}`
+
+        Transposition for matrix multiplied with scalar:
+        `\\left(c A\\right)^{T} = c A^{T}`
+
+        References
+        ==========
+
+        .. [1] https://en.wikipedia.org/wiki/Transpose
+        """
+        coeff, matrices = self.as_coeff_matrices()
+        return MatMul(
+            coeff, *[transpose(arg) for arg in matrices[::-1]]).doit()
+
+    def _eval_adjoint(self):
+        return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()
+
+    def _eval_trace(self):
+        factor, mmul = self.as_coeff_mmul()
+        if factor != 1:
+            from .trace import trace
+            return factor * trace(mmul.doit())
+        else:
+            raise NotImplementedError("Can't simplify any further")
+
+    def _eval_determinant(self):
+        from sympy.matrices.expressions.determinant import Determinant
+        factor, matrices = self.as_coeff_matrices()
+        square_matrices = only_squares(*matrices)
+        return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))
+
+    def _eval_inverse(self):
+        if all(arg.is_square for arg in self.args if isinstance(arg, MatrixExpr)):
+            return MatMul(*(
+                arg.inverse() if isinstance(arg, MatrixExpr) else arg**-1
+                    for arg in self.args[::-1]
+                )
+            ).doit()
+        return Inverse(self)
+
+    def doit(self, **hints):
+        deep = hints.get('deep', True)
+        if deep:
+            args = tuple(arg.doit(**hints) for arg in self.args)
+        else:
+            args = self.args
+
+        # treat scalar*MatrixSymbol or scalar*MatPow separately
+        expr = canonicalize(MatMul(*args))
+        return expr
+
+    # Needed for partial compatibility with Mul
+    def args_cnc(self, cset=False, warn=True, **kwargs):
+        coeff_c = [x for x in self.args if x.is_commutative]
+        coeff_nc = [x for x in self.args if not x.is_commutative]
+        if cset:
+            clen = len(coeff_c)
+            coeff_c = set(coeff_c)
+            if clen and warn and len(coeff_c) != clen:
+                raise ValueError('repeated commutative arguments: %s' %
+                                 [ci for ci in coeff_c if list(self.args).count(ci) > 1])
+        return [coeff_c, coeff_nc]
+
+    def _eval_derivative_matrix_lines(self, x):
+        from .transpose import Transpose
+        with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
+        lines = []
+        for ind in with_x_ind:
+            left_args = self.args[:ind]
+            right_args = self.args[ind+1:]
+
+            if right_args:
+                right_mat = MatMul.fromiter(right_args)
+            else:
+                right_mat = Identity(self.shape[1])
+            if left_args:
+                left_rev = MatMul.fromiter([Transpose(i).doit() if i.is_Matrix else i for i in reversed(left_args)])
+            else:
+                left_rev = Identity(self.shape[0])
+
+            d = self.args[ind]._eval_derivative_matrix_lines(x)
+            for i in d:
+                i.append_first(left_rev)
+                i.append_second(right_mat)
+                lines.append(i)
+
+        return lines
+
+mul.register_handlerclass((Mul, MatMul), MatMul)
+
+
+# Rules
+def newmul(*args):
+    if args[0] == 1:
+        args = args[1:]
+    return new(MatMul, *args)
+
+def any_zeros(mul):
+    if any(arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
+                       for arg in mul.args):
+        matrices = [arg for arg in mul.args if arg.is_Matrix]
+        return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
+    return mul
+
+def merge_explicit(matmul):
+    """ Merge explicit MatrixBase arguments
+
+    >>> from sympy import MatrixSymbol, Matrix, MatMul, pprint
+    >>> from sympy.matrices.expressions.matmul import merge_explicit
+    >>> A = MatrixSymbol('A', 2, 2)
+    >>> B = Matrix([[1, 1], [1, 1]])
+    >>> C = Matrix([[1, 2], [3, 4]])
+    >>> X = MatMul(A, B, C)
+    >>> pprint(X)
+      [1  1] [1  2]
+    A*[    ]*[    ]
+      [1  1] [3  4]
+    >>> pprint(merge_explicit(X))
+      [4  6]
+    A*[    ]
+      [4  6]
+
+    >>> X = MatMul(B, A, C)
+    >>> pprint(X)
+    [1  1]   [1  2]
+    [    ]*A*[    ]
+    [1  1]   [3  4]
+    >>> pprint(merge_explicit(X))
+    [1  1]   [1  2]
+    [    ]*A*[    ]
+    [1  1]   [3  4]
+    """
+    if not any(isinstance(arg, MatrixBase) for arg in matmul.args):
+        return matmul
+    newargs = []
+    last = matmul.args[0]
+    for arg in matmul.args[1:]:
+        if isinstance(arg, (MatrixBase, Number)) and isinstance(last, (MatrixBase, Number)):
+            last = last * arg
+        else:
+            newargs.append(last)
+            last = arg
+    newargs.append(last)
+
+    return MatMul(*newargs)
+
+def remove_ids(mul):
+    """ Remove Identities from a MatMul
+
+    This is a modified version of sympy.strategies.rm_id.
+    This is necesssary because MatMul may contain both MatrixExprs and Exprs
+    as args.
+
+    See Also
+    ========
+
+    sympy.strategies.rm_id
+    """
+    # Separate Exprs from MatrixExprs in args
+    factor, mmul = mul.as_coeff_mmul()
+    # Apply standard rm_id for MatMuls
+    result = rm_id(lambda x: x.is_Identity is True)(mmul)
+    if result != mmul:
+        return newmul(factor, *result.args)  # Recombine and return
+    else:
+        return mul
+
+def factor_in_front(mul):
+    factor, matrices = mul.as_coeff_matrices()
+    if factor != 1:
+        return newmul(factor, *matrices)
+    return mul
+
+def combine_powers(mul):
+    r"""Combine consecutive powers with the same base into one, e.g.
+    $$A \times A^2 \Rightarrow A^3$$
+
+    This also cancels out the possible matrix inverses using the
+    knowledgebase of :class:`~.Inverse`, e.g.,
+    $$ Y \times X \times X^{-1} \Rightarrow Y $$
+    """
+    factor, args = mul.as_coeff_matrices()
+    new_args = [args[0]]
+
+    for i in range(1, len(args)):
+        A = new_args[-1]
+        B = args[i]
+
+        if isinstance(B, Inverse) and isinstance(B.arg, MatMul):
+            Bargs = B.arg.args
+            l = len(Bargs)
+            if list(Bargs) == new_args[-l:]:
+                new_args = new_args[:-l] + [Identity(B.shape[0])]
+                continue
+
+        if isinstance(A, Inverse) and isinstance(A.arg, MatMul):
+            Aargs = A.arg.args
+            l = len(Aargs)
+            if list(Aargs) == args[i:i+l]:
+                identity = Identity(A.shape[0])
+                new_args[-1] = identity
+                for j in range(i, i+l):
+                    args[j] = identity
+                continue
+
+        if A.is_square == False or B.is_square == False:
+            new_args.append(B)
+            continue
+
+        if isinstance(A, MatPow):
+            A_base, A_exp = A.args
+        else:
+            A_base, A_exp = A, S.One
+
+        if isinstance(B, MatPow):
+            B_base, B_exp = B.args
+        else:
+            B_base, B_exp = B, S.One
+
+        if A_base == B_base:
+            new_exp = A_exp + B_exp
+            new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
+            continue
+        elif not isinstance(B_base, MatrixBase):
+            try:
+                B_base_inv = B_base.inverse()
+            except NonInvertibleMatrixError:
+                B_base_inv = None
+            if B_base_inv is not None and A_base == B_base_inv:
+                new_exp = A_exp - B_exp
+                new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
+                continue
+        new_args.append(B)
+
+    return newmul(factor, *new_args)
+
+def combine_permutations(mul):
+    """Refine products of permutation matrices as the products of cycles.
+    """
+    args = mul.args
+    l = len(args)
+    if l < 2:
+        return mul
+
+    result = [args[0]]
+    for i in range(1, l):
+        A = result[-1]
+        B = args[i]
+        if isinstance(A, PermutationMatrix) and \
+            isinstance(B, PermutationMatrix):
+            cycle_1 = A.args[0]
+            cycle_2 = B.args[0]
+            result[-1] = PermutationMatrix(cycle_1 * cycle_2)
+        else:
+            result.append(B)
+
+    return MatMul(*result)
+
+def combine_one_matrices(mul):
+    """
+    Combine products of OneMatrix
+
+    e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4)
+    """
+    factor, args = mul.as_coeff_matrices()
+    new_args = [args[0]]
+
+    for B in args[1:]:
+        A = new_args[-1]
+        if not isinstance(A, OneMatrix) or not isinstance(B, OneMatrix):
+            new_args.append(B)
+            continue
+        new_args.pop()
+        new_args.append(OneMatrix(A.shape[0], B.shape[1]))
+        factor *= A.shape[1]
+
+    return newmul(factor, *new_args)
+
+def distribute_monom(mul):
+    """
+    Simplify MatMul expressions but distributing
+    rational term to MatMul.
+
+    e.g. 2*(A+B) -> 2*A + 2*B
+    """
+    args = mul.args
+    if len(args) == 2:
+        from .matadd import MatAdd
+        if args[0].is_MatAdd and args[1].is_Rational:
+            return MatAdd(*[MatMul(mat, args[1]).doit() for mat in args[0].args])
+        if args[1].is_MatAdd and args[0].is_Rational:
+            return MatAdd(*[MatMul(args[0], mat).doit() for mat in args[1].args])
+    return mul
+
+rules = (
+    distribute_monom, any_zeros, remove_ids, combine_one_matrices, combine_powers, unpack, rm_id(lambda x: x == 1),
+    merge_explicit, factor_in_front, flatten, combine_permutations)
+
+canonicalize = exhaust(typed({MatMul: do_one(*rules)}))
+
+def only_squares(*matrices):
+    """factor matrices only if they are square"""
+    if matrices[0].rows != matrices[-1].cols:
+        raise RuntimeError("Invalid matrices being multiplied")
+    out = []
+    start = 0
+    for i, M in enumerate(matrices):
+        if M.cols == matrices[start].rows:
+            out.append(MatMul(*matrices[start:i+1]).doit())
+            start = i+1
+    return out
+
+
+def refine_MatMul(expr, assumptions):
+    """
+    >>> from sympy import MatrixSymbol, Q, assuming, refine
+    >>> X = MatrixSymbol('X', 2, 2)
+    >>> expr = X * X.T
+    >>> print(expr)
+    X*X.T
+    >>> with assuming(Q.orthogonal(X)):
+    ...     print(refine(expr))
+    I
+    """
+    newargs = []
+    exprargs = []
+
+    for args in expr.args:
+        if args.is_Matrix:
+            exprargs.append(args)
+        else:
+            newargs.append(args)
+
+    last = exprargs[0]
+    for arg in exprargs[1:]:
+        if arg == last.T and ask(Q.orthogonal(arg), assumptions):
+            last = Identity(arg.shape[0])
+        elif arg == last.conjugate() and ask(Q.unitary(arg), assumptions):
+            last = Identity(arg.shape[0])
+        else:
+            newargs.append(last)
+            last = arg
+    newargs.append(last)
+
+    return MatMul(*newargs)
+
+
+handlers_dict['MatMul'] = refine_MatMul

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/matpow.py

@@ -0,0 +1,142 @@
+from .matexpr import MatrixExpr
+from .special import Identity
+from sympy.core import S
+from sympy.core.expr import ExprBuilder
+from sympy.core.cache import cacheit
+from sympy.core.power import Pow
+from sympy.core.sympify import _sympify
+from sympy.matrices import MatrixBase
+from sympy.matrices.common import NonSquareMatrixError
+
+
+class MatPow(MatrixExpr):
+    def __new__(cls, base, exp, evaluate=False, **options):
+        base = _sympify(base)
+        if not base.is_Matrix:
+            raise TypeError("MatPow base should be a matrix")
+
+        if base.is_square is False:
+            raise NonSquareMatrixError("Power of non-square matrix %s" % base)
+
+        exp = _sympify(exp)
+        obj = super().__new__(cls, base, exp)
+
+        if evaluate:
+            obj = obj.doit(deep=False)
+
+        return obj
+
+    @property
+    def base(self):
+        return self.args[0]
+
+    @property
+    def exp(self):
+        return self.args[1]
+
+    @property
+    def shape(self):
+        return self.base.shape
+
+    @cacheit
+    def _get_explicit_matrix(self):
+        return self.base.as_explicit()**self.exp
+
+    def _entry(self, i, j, **kwargs):
+        from sympy.matrices.expressions import MatMul
+        A = self.doit()
+        if isinstance(A, MatPow):
+            # We still have a MatPow, make an explicit MatMul out of it.
+            if A.exp.is_Integer and A.exp.is_positive:
+                A = MatMul(*[A.base for k in range(A.exp)])
+            elif not self._is_shape_symbolic():
+                return A._get_explicit_matrix()[i, j]
+            else:
+                # Leave the expression unevaluated:
+                from sympy.matrices.expressions.matexpr import MatrixElement
+                return MatrixElement(self, i, j)
+        return A[i, j]
+
+    def doit(self, **hints):
+        if hints.get('deep', True):
+            base, exp = (arg.doit(**hints) for arg in self.args)
+        else:
+            base, exp = self.args
+
+        # combine all powers, e.g. (A ** 2) ** 3 -> A ** 6
+        while isinstance(base, MatPow):
+            exp *= base.args[1]
+            base = base.args[0]
+
+        if isinstance(base, MatrixBase):
+            # Delegate
+            return base ** exp
+
+        # Handle simple cases so that _eval_power() in MatrixExpr sub-classes can ignore them
+        if exp == S.One:
+            return base
+        if exp == S.Zero:
+            return Identity(base.rows)
+        if exp == S.NegativeOne:
+            from sympy.matrices.expressions import Inverse
+            return Inverse(base).doit(**hints)
+
+        eval_power = getattr(base, '_eval_power', None)
+        if eval_power is not None:
+            return eval_power(exp)
+
+        return MatPow(base, exp)
+
+    def _eval_transpose(self):
+        base, exp = self.args
+        return MatPow(base.T, exp)
+
+    def _eval_derivative(self, x):
+        return Pow._eval_derivative(self, x)
+
+    def _eval_derivative_matrix_lines(self, x):
+        from sympy.tensor.array.expressions.array_expressions import ArrayContraction
+        from ...tensor.array.expressions.array_expressions import ArrayTensorProduct
+        from .matmul import MatMul
+        from .inverse import Inverse
+        exp = self.exp
+        if self.base.shape == (1, 1) and not exp.has(x):
+            lr = self.base._eval_derivative_matrix_lines(x)
+            for i in lr:
+                subexpr = ExprBuilder(
+                    ArrayContraction,
+                    [
+                        ExprBuilder(
+                            ArrayTensorProduct,
+                            [
+                                Identity(1),
+                                i._lines[0],
+                                exp*self.base**(exp-1),
+                                i._lines[1],
+                                Identity(1),
+                            ]
+                        ),
+                        (0, 3, 4), (5, 7, 8)
+                    ],
+                    validator=ArrayContraction._validate
+                )
+                i._first_pointer_parent = subexpr.args[0].args
+                i._first_pointer_index = 0
+                i._second_pointer_parent = subexpr.args[0].args
+                i._second_pointer_index = 4
+                i._lines = [subexpr]
+            return lr
+        if (exp > 0) == True:
+            newexpr = MatMul.fromiter([self.base for i in range(exp)])
+        elif (exp == -1) == True:
+            return Inverse(self.base)._eval_derivative_matrix_lines(x)
+        elif (exp < 0) == True:
+            newexpr = MatMul.fromiter([Inverse(self.base) for i in range(-exp)])
+        elif (exp == 0) == True:
+            return self.doit()._eval_derivative_matrix_lines(x)
+        else:
+            raise NotImplementedError("cannot evaluate %s derived by %s" % (self, x))
+        return newexpr._eval_derivative_matrix_lines(x)
+
+    def _eval_inverse(self):
+        return MatPow(self.base, -self.exp)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/permutation.py

@@ -0,0 +1,303 @@
+from sympy.core import S
+from sympy.core.sympify import _sympify
+from sympy.functions import KroneckerDelta
+
+from .matexpr import MatrixExpr
+from .special import ZeroMatrix, Identity, OneMatrix
+
+
+class PermutationMatrix(MatrixExpr):
+    """A Permutation Matrix
+
+    Parameters
+    ==========
+
+    perm : Permutation
+        The permutation the matrix uses.
+
+        The size of the permutation determines the matrix size.
+
+        See the documentation of
+        :class:`sympy.combinatorics.permutations.Permutation` for
+        the further information of how to create a permutation object.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, PermutationMatrix
+    >>> from sympy.combinatorics import Permutation
+
+    Creating a permutation matrix:
+
+    >>> p = Permutation(1, 2, 0)
+    >>> P = PermutationMatrix(p)
+    >>> P = P.as_explicit()
+    >>> P
+    Matrix([
+    [0, 1, 0],
+    [0, 0, 1],
+    [1, 0, 0]])
+
+    Permuting a matrix row and column:
+
+    >>> M = Matrix([0, 1, 2])
+    >>> Matrix(P*M)
+    Matrix([
+    [1],
+    [2],
+    [0]])
+
+    >>> Matrix(M.T*P)
+    Matrix([[2, 0, 1]])
+
+    See Also
+    ========
+
+    sympy.combinatorics.permutations.Permutation
+    """
+
+    def __new__(cls, perm):
+        from sympy.combinatorics.permutations import Permutation
+
+        perm = _sympify(perm)
+        if not isinstance(perm, Permutation):
+            raise ValueError(
+                "{} must be a SymPy Permutation instance.".format(perm))
+
+        return super().__new__(cls, perm)
+
+    @property
+    def shape(self):
+        size = self.args[0].size
+        return (size, size)
+
+    @property
+    def is_Identity(self):
+        return self.args[0].is_Identity
+
+    def doit(self, **hints):
+        if self.is_Identity:
+            return Identity(self.rows)
+        return self
+
+    def _entry(self, i, j, **kwargs):
+        perm = self.args[0]
+        return KroneckerDelta(perm.apply(i), j)
+
+    def _eval_power(self, exp):
+        return PermutationMatrix(self.args[0] ** exp).doit()
+
+    def _eval_inverse(self):
+        return PermutationMatrix(self.args[0] ** -1)
+
+    _eval_transpose = _eval_adjoint = _eval_inverse
+
+    def _eval_determinant(self):
+        sign = self.args[0].signature()
+        if sign == 1:
+            return S.One
+        elif sign == -1:
+            return S.NegativeOne
+        raise NotImplementedError
+
+    def _eval_rewrite_as_BlockDiagMatrix(self, *args, **kwargs):
+        from sympy.combinatorics.permutations import Permutation
+        from .blockmatrix import BlockDiagMatrix
+
+        perm = self.args[0]
+        full_cyclic_form = perm.full_cyclic_form
+
+        cycles_picks = []
+
+        # Stage 1. Decompose the cycles into the blockable form.
+        a, b, c = 0, 0, 0
+        flag = False
+        for cycle in full_cyclic_form:
+            l = len(cycle)
+            m = max(cycle)
+
+            if not flag:
+                if m + 1 > a + l:
+                    flag = True
+                    temp = [cycle]
+                    b = m
+                    c = l
+                else:
+                    cycles_picks.append([cycle])
+                    a += l
+
+            else:
+                if m > b:
+                    if m + 1 == a + c + l:
+                        temp.append(cycle)
+                        cycles_picks.append(temp)
+                        flag = False
+                        a = m+1
+                    else:
+                        b = m
+                        temp.append(cycle)
+                        c += l
+                else:
+                    if b + 1 == a + c + l:
+                        temp.append(cycle)
+                        cycles_picks.append(temp)
+                        flag = False
+                        a = b+1
+                    else:
+                        temp.append(cycle)
+                        c += l
+
+        # Stage 2. Normalize each decomposed cycles and build matrix.
+        p = 0
+        args = []
+        for pick in cycles_picks:
+            new_cycles = []
+            l = 0
+            for cycle in pick:
+                new_cycle = [i - p for i in cycle]
+                new_cycles.append(new_cycle)
+                l += len(cycle)
+            p += l
+            perm = Permutation(new_cycles)
+            mat = PermutationMatrix(perm)
+            args.append(mat)
+
+        return BlockDiagMatrix(*args)
+
+
+class MatrixPermute(MatrixExpr):
+    r"""Symbolic representation for permuting matrix rows or columns.
+
+    Parameters
+    ==========
+
+    perm : Permutation, PermutationMatrix
+        The permutation to use for permuting the matrix.
+        The permutation can be resized to the suitable one,
+
+    axis : 0 or 1
+        The axis to permute alongside.
+        If `0`, it will permute the matrix rows.
+        If `1`, it will permute the matrix columns.
+
+    Notes
+    =====
+
+    This follows the same notation used in
+    :meth:`sympy.matrices.common.MatrixCommon.permute`.
+
+    Examples
+    ========
+
+    >>> from sympy import Matrix, MatrixPermute
+    >>> from sympy.combinatorics import Permutation
+
+    Permuting the matrix rows:
+
+    >>> p = Permutation(1, 2, 0)
+    >>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    >>> B = MatrixPermute(A, p, axis=0)
+    >>> B.as_explicit()
+    Matrix([
+    [4, 5, 6],
+    [7, 8, 9],
+    [1, 2, 3]])
+
+    Permuting the matrix columns:
+
+    >>> B = MatrixPermute(A, p, axis=1)
+    >>> B.as_explicit()
+    Matrix([
+    [2, 3, 1],
+    [5, 6, 4],
+    [8, 9, 7]])
+
+    See Also
+    ========
+
+    sympy.matrices.common.MatrixCommon.permute
+    """
+    def __new__(cls, mat, perm, axis=S.Zero):
+        from sympy.combinatorics.permutations import Permutation
+
+        mat = _sympify(mat)
+        if not mat.is_Matrix:
+            raise ValueError(
+                "{} must be a SymPy matrix instance.".format(perm))
+
+        perm = _sympify(perm)
+        if isinstance(perm, PermutationMatrix):
+            perm = perm.args[0]
+
+        if not isinstance(perm, Permutation):
+            raise ValueError(
+                "{} must be a SymPy Permutation or a PermutationMatrix " \
+                "instance".format(perm))
+
+        axis = _sympify(axis)
+        if axis not in (0, 1):
+            raise ValueError("The axis must be 0 or 1.")
+
+        mat_size = mat.shape[axis]
+        if mat_size != perm.size:
+            try:
+                perm = perm.resize(mat_size)
+            except ValueError:
+                raise ValueError(
+                    "Size does not match between the permutation {} "
+                    "and the matrix {} threaded over the axis {} "
+                    "and cannot be converted."
+                    .format(perm, mat, axis))
+
+        return super().__new__(cls, mat, perm, axis)
+
+    def doit(self, deep=True, **hints):
+        mat, perm, axis = self.args
+
+        if deep:
+            mat = mat.doit(deep=deep, **hints)
+            perm = perm.doit(deep=deep, **hints)
+
+        if perm.is_Identity:
+            return mat
+
+        if mat.is_Identity:
+            if axis is S.Zero:
+                return PermutationMatrix(perm)
+            elif axis is S.One:
+                return PermutationMatrix(perm**-1)
+
+        if isinstance(mat, (ZeroMatrix, OneMatrix)):
+            return mat
+
+        if isinstance(mat, MatrixPermute) and mat.args[2] == axis:
+            return MatrixPermute(mat.args[0], perm * mat.args[1], axis)
+
+        return self
+
+    @property
+    def shape(self):
+        return self.args[0].shape
+
+    def _entry(self, i, j, **kwargs):
+        mat, perm, axis = self.args
+
+        if axis == 0:
+            return mat[perm.apply(i), j]
+        elif axis == 1:
+            return mat[i, perm.apply(j)]
+
+    def _eval_rewrite_as_MatMul(self, *args, **kwargs):
+        from .matmul import MatMul
+
+        mat, perm, axis = self.args
+
+        deep = kwargs.get("deep", True)
+
+        if deep:
+            mat = mat.rewrite(MatMul)
+
+        if axis == 0:
+            return MatMul(PermutationMatrix(perm), mat)
+        elif axis == 1:
+            return MatMul(mat, PermutationMatrix(perm**-1))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/sets.py

@@ -0,0 +1,67 @@
+from sympy.core.assumptions import check_assumptions
+from sympy.core.logic import fuzzy_and
+from sympy.core.sympify import _sympify
+from sympy.matrices.common import MatrixKind
+from sympy.sets.sets import Set, SetKind
+from sympy.core.kind import NumberKind
+from .matexpr import MatrixExpr
+
+
+class MatrixSet(Set):
+    """
+    MatrixSet represents the set of matrices with ``shape = (n, m)`` over the
+    given set.
+
+    Examples
+    ========
+
+    >>> from sympy.matrices import MatrixSet
+    >>> from sympy import S, I, Matrix
+    >>> M = MatrixSet(2, 2, set=S.Reals)
+    >>> X = Matrix([[1, 2], [3, 4]])
+    >>> X in M
+    True
+    >>> X = Matrix([[1, 2], [I, 4]])
+    >>> X in M
+    False
+
+    """
+    is_empty = False
+
+    def __new__(cls, n, m, set):
+        n, m, set = _sympify(n), _sympify(m), _sympify(set)
+        cls._check_dim(n)
+        cls._check_dim(m)
+        if not isinstance(set, Set):
+            raise TypeError("{} should be an instance of Set.".format(set))
+        return Set.__new__(cls, n, m, set)
+
+    @property
+    def shape(self):
+        return self.args[:2]
+
+    @property
+    def set(self):
+        return self.args[2]
+
+    def _contains(self, other):
+        if not isinstance(other, MatrixExpr):
+            raise TypeError("{} should be an instance of MatrixExpr.".format(other))
+        if other.shape != self.shape:
+            are_symbolic = any(_sympify(x).is_Symbol for x in other.shape + self.shape)
+            if are_symbolic:
+                return None
+            return False
+        return fuzzy_and(self.set.contains(x) for x in other)
+
+    @classmethod
+    def _check_dim(cls, dim):
+        """Helper function to check invalid matrix dimensions"""
+        ok = check_assumptions(dim, integer=True, nonnegative=True)
+        if ok is False:
+            raise ValueError(
+                "The dimension specification {} should be "
+                "a nonnegative integer.".format(dim))
+
+    def _kind(self):
+        return SetKind(MatrixKind(NumberKind))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/slice.py

@@ -0,0 +1,114 @@
+from sympy.matrices.expressions.matexpr import MatrixExpr
+from sympy.core.basic import Basic
+from sympy.core.containers import Tuple
+from sympy.functions.elementary.integers import floor
+
+def normalize(i, parentsize):
+    if isinstance(i, slice):
+        i = (i.start, i.stop, i.step)
+    if not isinstance(i, (tuple, list, Tuple)):
+        if (i < 0) == True:
+            i += parentsize
+        i = (i, i+1, 1)
+    i = list(i)
+    if len(i) == 2:
+        i.append(1)
+    start, stop, step = i
+    start = start or 0
+    if stop is None:
+        stop = parentsize
+    if (start < 0) == True:
+        start += parentsize
+    if (stop < 0) == True:
+        stop += parentsize
+    step = step or 1
+
+    if ((stop - start) * step < 1) == True:
+        raise IndexError()
+
+    return (start, stop, step)
+
+class MatrixSlice(MatrixExpr):
+    """ A MatrixSlice of a Matrix Expression
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSlice, ImmutableMatrix
+    >>> M = ImmutableMatrix(4, 4, range(16))
+    >>> M
+    Matrix([
+    [ 0,  1,  2,  3],
+    [ 4,  5,  6,  7],
+    [ 8,  9, 10, 11],
+    [12, 13, 14, 15]])
+
+    >>> B = MatrixSlice(M, (0, 2), (2, 4))
+    >>> ImmutableMatrix(B)
+    Matrix([
+    [2, 3],
+    [6, 7]])
+    """
+    parent = property(lambda self: self.args[0])
+    rowslice = property(lambda self: self.args[1])
+    colslice = property(lambda self: self.args[2])
+
+    def __new__(cls, parent, rowslice, colslice):
+        rowslice = normalize(rowslice, parent.shape[0])
+        colslice = normalize(colslice, parent.shape[1])
+        if not (len(rowslice) == len(colslice) == 3):
+            raise IndexError()
+        if ((0 > rowslice[0]) == True or
+            (parent.shape[0] < rowslice[1]) == True or
+            (0 > colslice[0]) == True or
+            (parent.shape[1] < colslice[1]) == True):
+            raise IndexError()
+        if isinstance(parent, MatrixSlice):
+            return mat_slice_of_slice(parent, rowslice, colslice)
+        return Basic.__new__(cls, parent, Tuple(*rowslice), Tuple(*colslice))
+
+    @property
+    def shape(self):
+        rows = self.rowslice[1] - self.rowslice[0]
+        rows = rows if self.rowslice[2] == 1 else floor(rows/self.rowslice[2])
+        cols = self.colslice[1] - self.colslice[0]
+        cols = cols if self.colslice[2] == 1 else floor(cols/self.colslice[2])
+        return rows, cols
+
+    def _entry(self, i, j, **kwargs):
+        return self.parent._entry(i*self.rowslice[2] + self.rowslice[0],
+                                  j*self.colslice[2] + self.colslice[0],
+                                  **kwargs)
+
+    @property
+    def on_diag(self):
+        return self.rowslice == self.colslice
+
+
+def slice_of_slice(s, t):
+    start1, stop1, step1 = s
+    start2, stop2, step2 = t
+
+    start = start1 + start2*step1
+    step = step1 * step2
+    stop = start1 + step1*stop2
+
+    if stop > stop1:
+        raise IndexError()
+
+    return start, stop, step
+
+
+def mat_slice_of_slice(parent, rowslice, colslice):
+    """ Collapse nested matrix slices
+
+    >>> from sympy import MatrixSymbol
+    >>> X = MatrixSymbol('X', 10, 10)
+    >>> X[:, 1:5][5:8, :]
+    X[5:8, 1:5]
+    >>> X[1:9:2, 2:6][1:3, 2]
+    X[3:7:2, 4:5]
+    """
+    row = slice_of_slice(parent.rowslice, rowslice)
+    col = slice_of_slice(parent.colslice, colslice)
+    return MatrixSlice(parent.parent, row, col)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/special.py

@@ -0,0 +1,299 @@
+from sympy.assumptions.ask import ask, Q
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.sympify import _sympify
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.matrices.common import NonInvertibleMatrixError
+from .matexpr import MatrixExpr
+
+
+class ZeroMatrix(MatrixExpr):
+    """The Matrix Zero 0 - additive identity
+
+    Examples
+    ========
+
+    >>> from sympy import MatrixSymbol, ZeroMatrix
+    >>> A = MatrixSymbol('A', 3, 5)
+    >>> Z = ZeroMatrix(3, 5)
+    >>> A + Z
+    A
+    >>> Z*A.T
+    0
+    """
+    is_ZeroMatrix = True
+
+    def __new__(cls, m, n):
+        m, n = _sympify(m), _sympify(n)
+        cls._check_dim(m)
+        cls._check_dim(n)
+
+        return super().__new__(cls, m, n)
+
+    @property
+    def shape(self):
+        return (self.args[0], self.args[1])
+
+    def _eval_power(self, exp):
+        # exp = -1, 0, 1 are already handled at this stage
+        if (exp < 0) == True:
+            raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
+        return self
+
+    def _eval_transpose(self):
+        return ZeroMatrix(self.cols, self.rows)
+
+    def _eval_adjoint(self):
+        return ZeroMatrix(self.cols, self.rows)
+
+    def _eval_trace(self):
+        return S.Zero
+
+    def _eval_determinant(self):
+        return S.Zero
+
+    def _eval_inverse(self):
+        raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
+
+    def _eval_as_real_imag(self):
+        return (self, self)
+
+    def _eval_conjugate(self):
+        return self
+
+    def _entry(self, i, j, **kwargs):
+        return S.Zero
+
+
+class GenericZeroMatrix(ZeroMatrix):
+    """
+    A zero matrix without a specified shape
+
+    This exists primarily so MatAdd() with no arguments can return something
+    meaningful.
+    """
+    def __new__(cls):
+        # super(ZeroMatrix, cls) instead of super(GenericZeroMatrix, cls)
+        # because ZeroMatrix.__new__ doesn't have the same signature
+        return super(ZeroMatrix, cls).__new__(cls)
+
+    @property
+    def rows(self):
+        raise TypeError("GenericZeroMatrix does not have a specified shape")
+
+    @property
+    def cols(self):
+        raise TypeError("GenericZeroMatrix does not have a specified shape")
+
+    @property
+    def shape(self):
+        raise TypeError("GenericZeroMatrix does not have a specified shape")
+
+    # Avoid Matrix.__eq__ which might call .shape
+    def __eq__(self, other):
+        return isinstance(other, GenericZeroMatrix)
+
+    def __ne__(self, other):
+        return not (self == other)
+
+    def __hash__(self):
+        return super().__hash__()
+
+
+
+class Identity(MatrixExpr):
+    """The Matrix Identity I - multiplicative identity
+
+    Examples
+    ========
+
+    >>> from sympy import Identity, MatrixSymbol
+    >>> A = MatrixSymbol('A', 3, 5)
+    >>> I = Identity(3)
+    >>> I*A
+    A
+    """
+
+    is_Identity = True
+
+    def __new__(cls, n):
+        n = _sympify(n)
+        cls._check_dim(n)
+
+        return super().__new__(cls, n)
+
+    @property
+    def rows(self):
+        return self.args[0]
+
+    @property
+    def cols(self):
+        return self.args[0]
+
+    @property
+    def shape(self):
+        return (self.args[0], self.args[0])
+
+    @property
+    def is_square(self):
+        return True
+
+    def _eval_transpose(self):
+        return self
+
+    def _eval_trace(self):
+        return self.rows
+
+    def _eval_inverse(self):
+        return self
+
+    def _eval_as_real_imag(self):
+        return (self, ZeroMatrix(*self.shape))
+
+    def _eval_conjugate(self):
+        return self
+
+    def _eval_adjoint(self):
+        return self
+
+    def _entry(self, i, j, **kwargs):
+        eq = Eq(i, j)
+        if eq is S.true:
+            return S.One
+        elif eq is S.false:
+            return S.Zero
+        return KroneckerDelta(i, j, (0, self.cols-1))
+
+    def _eval_determinant(self):
+        return S.One
+
+    def _eval_power(self, exp):
+        return self
+
+
+class GenericIdentity(Identity):
+    """
+    An identity matrix without a specified shape
+
+    This exists primarily so MatMul() with no arguments can return something
+    meaningful.
+    """
+    def __new__(cls):
+        # super(Identity, cls) instead of super(GenericIdentity, cls) because
+        # Identity.__new__ doesn't have the same signature
+        return super(Identity, cls).__new__(cls)
+
+    @property
+    def rows(self):
+        raise TypeError("GenericIdentity does not have a specified shape")
+
+    @property
+    def cols(self):
+        raise TypeError("GenericIdentity does not have a specified shape")
+
+    @property
+    def shape(self):
+        raise TypeError("GenericIdentity does not have a specified shape")
+
+    @property
+    def is_square(self):
+        return True
+
+    # Avoid Matrix.__eq__ which might call .shape
+    def __eq__(self, other):
+        return isinstance(other, GenericIdentity)
+
+    def __ne__(self, other):
+        return not (self == other)
+
+    def __hash__(self):
+        return super().__hash__()
+
+
+class OneMatrix(MatrixExpr):
+    """
+    Matrix whose all entries are ones.
+    """
+    def __new__(cls, m, n, evaluate=False):
+        m, n = _sympify(m), _sympify(n)
+        cls._check_dim(m)
+        cls._check_dim(n)
+
+        if evaluate:
+            condition = Eq(m, 1) & Eq(n, 1)
+            if condition == True:
+                return Identity(1)
+
+        obj = super().__new__(cls, m, n)
+        return obj
+
+    @property
+    def shape(self):
+        return self._args
+
+    @property
+    def is_Identity(self):
+        return self._is_1x1() == True
+
+    def as_explicit(self):
+        from sympy.matrices.immutable import ImmutableDenseMatrix
+        return ImmutableDenseMatrix.ones(*self.shape)
+
+    def doit(self, **hints):
+        args = self.args
+        if hints.get('deep', True):
+            args = [a.doit(**hints) for a in args]
+        return self.func(*args, evaluate=True)
+
+    def _eval_power(self, exp):
+        # exp = -1, 0, 1 are already handled at this stage
+        if self._is_1x1() == True:
+            return Identity(1)
+        if (exp < 0) == True:
+            raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
+        if ask(Q.integer(exp)):
+            return self.shape[0] ** (exp - 1) * OneMatrix(*self.shape)
+        return super()._eval_power(exp)
+
+    def _eval_transpose(self):
+        return OneMatrix(self.cols, self.rows)
+
+    def _eval_adjoint(self):
+        return OneMatrix(self.cols, self.rows)
+
+    def _eval_trace(self):
+        return S.One*self.rows
+
+    def _is_1x1(self):
+        """Returns true if the matrix is known to be 1x1"""
+        shape = self.shape
+        return Eq(shape[0], 1) & Eq(shape[1], 1)
+
+    def _eval_determinant(self):
+        condition = self._is_1x1()
+        if condition == True:
+            return S.One
+        elif condition == False:
+            return S.Zero
+        else:
+            from sympy.matrices.expressions.determinant import Determinant
+            return Determinant(self)
+
+    def _eval_inverse(self):
+        condition = self._is_1x1()
+        if condition == True:
+            return Identity(1)
+        elif condition == False:
+            raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
+        else:
+            from .inverse import Inverse
+            return Inverse(self)
+
+    def _eval_as_real_imag(self):
+        return (self, ZeroMatrix(*self.shape))
+
+    def _eval_conjugate(self):
+        return self
+
+    def _entry(self, i, j, **kwargs):
+        return S.One

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_adjoint.py

@@ -0,0 +1,34 @@
+from sympy.core import symbols, S
+from sympy.functions import adjoint, conjugate, transpose
+from sympy.matrices.expressions import MatrixSymbol, Adjoint, trace, Transpose
+from sympy.matrices import eye, Matrix
+
+n, m, l, k, p = symbols('n m l k p', integer=True)
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', m, l)
+C = MatrixSymbol('C', n, n)
+
+
+def test_adjoint():
+    Sq = MatrixSymbol('Sq', n, n)
+
+    assert Adjoint(A).shape == (m, n)
+    assert Adjoint(A*B).shape == (l, n)
+    assert adjoint(Adjoint(A)) == A
+    assert isinstance(Adjoint(Adjoint(A)), Adjoint)
+
+    assert conjugate(Adjoint(A)) == Transpose(A)
+    assert transpose(Adjoint(A)) == Adjoint(Transpose(A))
+
+    assert Adjoint(eye(3)).doit() == eye(3)
+
+    assert Adjoint(S(5)).doit() == S(5)
+
+    assert Adjoint(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3], [2, 4]])
+
+    assert adjoint(trace(Sq)) == conjugate(trace(Sq))
+    assert trace(adjoint(Sq)) == conjugate(trace(Sq))
+
+    assert Adjoint(Sq)[0, 1] == conjugate(Sq[1, 0])
+
+    assert Adjoint(A*B).doit() == Adjoint(B) * Adjoint(A)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_applyfunc.py

@@ -0,0 +1,118 @@
+from sympy.core.symbol import symbols, Dummy
+from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
+from sympy.core.function import Lambda
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.trigonometric import sin
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.matmul import MatMul
+from sympy.simplify.simplify import simplify
+
+
+X = MatrixSymbol("X", 3, 3)
+Y = MatrixSymbol("Y", 3, 3)
+
+k = symbols("k")
+Xk = MatrixSymbol("X", k, k)
+
+Xd = X.as_explicit()
+
+x, y, z, t = symbols("x y z t")
+
+
+def test_applyfunc_matrix():
+    x = Dummy('x')
+    double = Lambda(x, x**2)
+
+    expr = ElementwiseApplyFunction(double, Xd)
+    assert isinstance(expr, ElementwiseApplyFunction)
+    assert expr.doit() == Xd.applyfunc(lambda x: x**2)
+    assert expr.shape == (3, 3)
+    assert expr.func(*expr.args) == expr
+    assert simplify(expr) == expr
+    assert expr[0, 0] == double(Xd[0, 0])
+
+    expr = ElementwiseApplyFunction(double, X)
+    assert isinstance(expr, ElementwiseApplyFunction)
+    assert isinstance(expr.doit(), ElementwiseApplyFunction)
+    assert expr == X.applyfunc(double)
+    assert expr.func(*expr.args) == expr
+
+    expr = ElementwiseApplyFunction(exp, X*Y)
+    assert expr.expr == X*Y
+    assert expr.function.dummy_eq(Lambda(x, exp(x)))
+    assert expr.dummy_eq((X*Y).applyfunc(exp))
+    assert expr.func(*expr.args) == expr
+
+    assert isinstance(X*expr, MatMul)
+    assert (X*expr).shape == (3, 3)
+    Z = MatrixSymbol("Z", 2, 3)
+    assert (Z*expr).shape == (2, 3)
+
+    expr = ElementwiseApplyFunction(exp, Z.T)*ElementwiseApplyFunction(exp, Z)
+    assert expr.shape == (3, 3)
+    expr = ElementwiseApplyFunction(exp, Z)*ElementwiseApplyFunction(exp, Z.T)
+    assert expr.shape == (2, 2)
+
+    M = Matrix([[x, y], [z, t]])
+    expr = ElementwiseApplyFunction(sin, M)
+    assert isinstance(expr, ElementwiseApplyFunction)
+    assert expr.function.dummy_eq(Lambda(x, sin(x)))
+    assert expr.expr == M
+    assert expr.doit() == M.applyfunc(sin)
+    assert expr.doit() == Matrix([[sin(x), sin(y)], [sin(z), sin(t)]])
+    assert expr.func(*expr.args) == expr
+
+    expr = ElementwiseApplyFunction(double, Xk)
+    assert expr.doit() == expr
+    assert expr.subs(k, 2).shape == (2, 2)
+    assert (expr*expr).shape == (k, k)
+    M = MatrixSymbol("M", k, t)
+    expr2 = M.T*expr*M
+    assert isinstance(expr2, MatMul)
+    assert expr2.args[1] == expr
+    assert expr2.shape == (t, t)
+    expr3 = expr*M
+    assert expr3.shape == (k, t)
+
+    expr1 = ElementwiseApplyFunction(lambda x: x+1, Xk)
+    expr2 = ElementwiseApplyFunction(lambda x: x, Xk)
+    assert expr1 != expr2
+
+
+def test_applyfunc_entry():
+
+    af = X.applyfunc(sin)
+    assert af[0, 0] == sin(X[0, 0])
+
+    af = Xd.applyfunc(sin)
+    assert af[0, 0] == sin(X[0, 0])
+
+
+def test_applyfunc_as_explicit():
+
+    af = X.applyfunc(sin)
+    assert af.as_explicit() == Matrix([
+        [sin(X[0, 0]), sin(X[0, 1]), sin(X[0, 2])],
+        [sin(X[1, 0]), sin(X[1, 1]), sin(X[1, 2])],
+        [sin(X[2, 0]), sin(X[2, 1]), sin(X[2, 2])],
+    ])
+
+
+def test_applyfunc_transpose():
+
+    af = Xk.applyfunc(sin)
+    assert af.T.dummy_eq(Xk.T.applyfunc(sin))
+
+
+def test_applyfunc_shape_11_matrices():
+    M = MatrixSymbol("M", 1, 1)
+
+    double = Lambda(x, x*2)
+
+    expr = M.applyfunc(sin)
+    assert isinstance(expr, ElementwiseApplyFunction)
+
+    expr = M.applyfunc(double)
+    assert isinstance(expr, MatMul)
+    assert expr == 2*M

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_blockmatrix.py

@@ -0,0 +1,445 @@
+from sympy.matrices.expressions.trace import Trace
+from sympy.testing.pytest import raises, slow
+from sympy.matrices.expressions.blockmatrix import (
+    block_collapse, bc_matmul, bc_block_plus_ident, BlockDiagMatrix,
+    BlockMatrix, bc_dist, bc_matadd, bc_transpose, bc_inverse,
+    blockcut, reblock_2x2, deblock)
+from sympy.matrices.expressions import (MatrixSymbol, Identity,
+        Inverse, trace, Transpose, det, ZeroMatrix, OneMatrix)
+from sympy.matrices.common import NonInvertibleMatrixError
+from sympy.matrices import (
+    Matrix, ImmutableMatrix, ImmutableSparseMatrix)
+from sympy.core import Tuple, symbols, Expr, S
+from sympy.functions import transpose, im, re
+
+i, j, k, l, m, n, p = symbols('i:n, p', integer=True)
+A = MatrixSymbol('A', n, n)
+B = MatrixSymbol('B', n, n)
+C = MatrixSymbol('C', n, n)
+D = MatrixSymbol('D', n, n)
+G = MatrixSymbol('G', n, n)
+H = MatrixSymbol('H', n, n)
+b1 = BlockMatrix([[G, H]])
+b2 = BlockMatrix([[G], [H]])
+
+def test_bc_matmul():
+    assert bc_matmul(H*b1*b2*G) == BlockMatrix([[(H*G*G + H*H*H)*G]])
+
+def test_bc_matadd():
+    assert bc_matadd(BlockMatrix([[G, H]]) + BlockMatrix([[H, H]])) == \
+            BlockMatrix([[G+H, H+H]])
+
+def test_bc_transpose():
+    assert bc_transpose(Transpose(BlockMatrix([[A, B], [C, D]]))) == \
+            BlockMatrix([[A.T, C.T], [B.T, D.T]])
+
+def test_bc_dist_diag():
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', m, m)
+    C = MatrixSymbol('C', l, l)
+    X = BlockDiagMatrix(A, B, C)
+
+    assert bc_dist(X+X).equals(BlockDiagMatrix(2*A, 2*B, 2*C))
+
+def test_block_plus_ident():
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', n, m)
+    C = MatrixSymbol('C', m, n)
+    D = MatrixSymbol('D', m, m)
+    X = BlockMatrix([[A, B], [C, D]])
+    Z = MatrixSymbol('Z', n + m, n + m)
+    assert bc_block_plus_ident(X + Identity(m + n) + Z) == \
+            BlockDiagMatrix(Identity(n), Identity(m)) + X + Z
+
+def test_BlockMatrix():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', n, k)
+    C = MatrixSymbol('C', l, m)
+    D = MatrixSymbol('D', l, k)
+    M = MatrixSymbol('M', m + k, p)
+    N = MatrixSymbol('N', l + n, k + m)
+    X = BlockMatrix(Matrix([[A, B], [C, D]]))
+
+    assert X.__class__(*X.args) == X
+
+    # block_collapse does nothing on normal inputs
+    E = MatrixSymbol('E', n, m)
+    assert block_collapse(A + 2*E) == A + 2*E
+    F = MatrixSymbol('F', m, m)
+    assert block_collapse(E.T*A*F) == E.T*A*F
+
+    assert X.shape == (l + n, k + m)
+    assert X.blockshape == (2, 2)
+    assert transpose(X) == BlockMatrix(Matrix([[A.T, C.T], [B.T, D.T]]))
+    assert transpose(X).shape == X.shape[::-1]
+
+    # Test that BlockMatrices and MatrixSymbols can still mix
+    assert (X*M).is_MatMul
+    assert X._blockmul(M).is_MatMul
+    assert (X*M).shape == (n + l, p)
+    assert (X + N).is_MatAdd
+    assert X._blockadd(N).is_MatAdd
+    assert (X + N).shape == X.shape
+
+    E = MatrixSymbol('E', m, 1)
+    F = MatrixSymbol('F', k, 1)
+
+    Y = BlockMatrix(Matrix([[E], [F]]))
+
+    assert (X*Y).shape == (l + n, 1)
+    assert block_collapse(X*Y).blocks[0, 0] == A*E + B*F
+    assert block_collapse(X*Y).blocks[1, 0] == C*E + D*F
+
+    # block_collapse passes down into container objects, transposes, and inverse
+    assert block_collapse(transpose(X*Y)) == transpose(block_collapse(X*Y))
+    assert block_collapse(Tuple(X*Y, 2*X)) == (
+        block_collapse(X*Y), block_collapse(2*X))
+
+    # Make sure that MatrixSymbols will enter 1x1 BlockMatrix if it simplifies
+    Ab = BlockMatrix([[A]])
+    Z = MatrixSymbol('Z', *A.shape)
+    assert block_collapse(Ab + Z) == A + Z
+
+def test_block_collapse_explicit_matrices():
+    A = Matrix([[1, 2], [3, 4]])
+    assert block_collapse(BlockMatrix([[A]])) == A
+
+    A = ImmutableSparseMatrix([[1, 2], [3, 4]])
+    assert block_collapse(BlockMatrix([[A]])) == A
+
+def test_issue_17624():
+    a = MatrixSymbol("a", 2, 2)
+    z = ZeroMatrix(2, 2)
+    b = BlockMatrix([[a, z], [z, z]])
+    assert block_collapse(b * b) == BlockMatrix([[a**2, z], [z, z]])
+    assert block_collapse(b * b * b) == BlockMatrix([[a**3, z], [z, z]])
+
+def test_issue_18618():
+    A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    assert A == Matrix(BlockDiagMatrix(A))
+
+def test_BlockMatrix_trace():
+    A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
+    X = BlockMatrix([[A, B], [C, D]])
+    assert trace(X) == trace(A) + trace(D)
+    assert trace(BlockMatrix([ZeroMatrix(n, n)])) == 0
+
+def test_BlockMatrix_Determinant():
+    A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
+    X = BlockMatrix([[A, B], [C, D]])
+    from sympy.assumptions.ask import Q
+    from sympy.assumptions.assume import assuming
+    with assuming(Q.invertible(A)):
+        assert det(X) == det(A) * det(X.schur('A'))
+
+    assert isinstance(det(X), Expr)
+    assert det(BlockMatrix([A])) == det(A)
+    assert det(BlockMatrix([ZeroMatrix(n, n)])) == 0
+
+def test_squareBlockMatrix():
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', n, m)
+    C = MatrixSymbol('C', m, n)
+    D = MatrixSymbol('D', m, m)
+    X = BlockMatrix([[A, B], [C, D]])
+    Y = BlockMatrix([[A]])
+
+    assert X.is_square
+
+    Q = X + Identity(m + n)
+    assert (block_collapse(Q) ==
+        BlockMatrix([[A + Identity(n), B], [C, D + Identity(m)]]))
+
+    assert (X + MatrixSymbol('Q', n + m, n + m)).is_MatAdd
+    assert (X * MatrixSymbol('Q', n + m, n + m)).is_MatMul
+
+    assert block_collapse(Y.I) == A.I
+
+    assert isinstance(X.inverse(), Inverse)
+
+    assert not X.is_Identity
+
+    Z = BlockMatrix([[Identity(n), B], [C, D]])
+    assert not Z.is_Identity
+
+
+def test_BlockMatrix_2x2_inverse_symbolic():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', n, k - m)
+    C = MatrixSymbol('C', k - n, m)
+    D = MatrixSymbol('D', k - n, k - m)
+    X = BlockMatrix([[A, B], [C, D]])
+    assert X.is_square and X.shape == (k, k)
+    assert isinstance(block_collapse(X.I), Inverse)  # Can't invert when none of the blocks is square
+
+    # test code path where only A is invertible
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', n, m)
+    C = MatrixSymbol('C', m, n)
+    D = ZeroMatrix(m, m)
+    X = BlockMatrix([[A, B], [C, D]])
+    assert block_collapse(X.inverse()) == BlockMatrix([
+        [A.I + A.I * B * X.schur('A').I * C * A.I, -A.I * B * X.schur('A').I],
+        [-X.schur('A').I * C * A.I, X.schur('A').I],
+    ])
+
+    # test code path where only B is invertible
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', n, n)
+    C = ZeroMatrix(m, m)
+    D = MatrixSymbol('D', m, n)
+    X = BlockMatrix([[A, B], [C, D]])
+    assert block_collapse(X.inverse()) == BlockMatrix([
+        [-X.schur('B').I * D * B.I, X.schur('B').I],
+        [B.I + B.I * A * X.schur('B').I * D * B.I, -B.I * A * X.schur('B').I],
+    ])
+
+    # test code path where only C is invertible
+    A = MatrixSymbol('A', n, m)
+    B = ZeroMatrix(n, n)
+    C = MatrixSymbol('C', m, m)
+    D = MatrixSymbol('D', m, n)
+    X = BlockMatrix([[A, B], [C, D]])
+    assert block_collapse(X.inverse()) == BlockMatrix([
+        [-C.I * D * X.schur('C').I, C.I + C.I * D * X.schur('C').I * A * C.I],
+        [X.schur('C').I, -X.schur('C').I * A * C.I],
+    ])
+
+    # test code path where only D is invertible
+    A = ZeroMatrix(n, n)
+    B = MatrixSymbol('B', n, m)
+    C = MatrixSymbol('C', m, n)
+    D = MatrixSymbol('D', m, m)
+    X = BlockMatrix([[A, B], [C, D]])
+    assert block_collapse(X.inverse()) == BlockMatrix([
+        [X.schur('D').I, -X.schur('D').I * B * D.I],
+        [-D.I * C * X.schur('D').I, D.I + D.I * C * X.schur('D').I * B * D.I],
+    ])
+
+
+def test_BlockMatrix_2x2_inverse_numeric():
+    """Test 2x2 block matrix inversion numerically for all 4 formulas"""
+    M = Matrix([[1, 2], [3, 4]])
+    # rank deficient matrices that have full rank when two of them combined
+    D1 = Matrix([[1, 2], [2, 4]])
+    D2 = Matrix([[1, 3], [3, 9]])
+    D3 = Matrix([[1, 4], [4, 16]])
+    assert D1.rank() == D2.rank() == D3.rank() == 1
+    assert (D1 + D2).rank() == (D2 + D3).rank() == (D3 + D1).rank() == 2
+
+    # Only A is invertible
+    K = BlockMatrix([[M, D1], [D2, D3]])
+    assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
+    # Only B is invertible
+    K = BlockMatrix([[D1, M], [D2, D3]])
+    assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
+    # Only C is invertible
+    K = BlockMatrix([[D1, D2], [M, D3]])
+    assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
+    # Only D is invertible
+    K = BlockMatrix([[D1, D2], [D3, M]])
+    assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
+
+
+@slow
+def test_BlockMatrix_3x3_symbolic():
+    # Only test one of these, instead of all permutations, because it's slow
+    rowblocksizes = (n, m, k)
+    colblocksizes = (m, k, n)
+    K = BlockMatrix([
+        [MatrixSymbol('M%s%s' % (rows, cols), rows, cols) for cols in colblocksizes]
+        for rows in rowblocksizes
+    ])
+    collapse = block_collapse(K.I)
+    assert isinstance(collapse, BlockMatrix)
+
+
+def test_BlockDiagMatrix():
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', m, m)
+    C = MatrixSymbol('C', l, l)
+    M = MatrixSymbol('M', n + m + l, n + m + l)
+
+    X = BlockDiagMatrix(A, B, C)
+    Y = BlockDiagMatrix(A, 2*B, 3*C)
+
+    assert X.blocks[1, 1] == B
+    assert X.shape == (n + m + l, n + m + l)
+    assert all(X.blocks[i, j].is_ZeroMatrix if i != j else X.blocks[i, j] in [A, B, C]
+            for i in range(3) for j in range(3))
+    assert X.__class__(*X.args) == X
+    assert X.get_diag_blocks() == (A, B, C)
+
+    assert isinstance(block_collapse(X.I * X), Identity)
+
+    assert bc_matmul(X*X) == BlockDiagMatrix(A*A, B*B, C*C)
+    assert block_collapse(X*X) == BlockDiagMatrix(A*A, B*B, C*C)
+    #XXX: should be == ??
+    assert block_collapse(X + X).equals(BlockDiagMatrix(2*A, 2*B, 2*C))
+    assert block_collapse(X*Y) == BlockDiagMatrix(A*A, 2*B*B, 3*C*C)
+    assert block_collapse(X + Y) == BlockDiagMatrix(2*A, 3*B, 4*C)
+
+    # Ensure that BlockDiagMatrices can still interact with normal MatrixExprs
+    assert (X*(2*M)).is_MatMul
+    assert (X + (2*M)).is_MatAdd
+
+    assert (X._blockmul(M)).is_MatMul
+    assert (X._blockadd(M)).is_MatAdd
+
+def test_BlockDiagMatrix_nonsquare():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', k, l)
+    X = BlockDiagMatrix(A, B)
+    assert X.shape == (n + k, m + l)
+    assert X.shape == (n + k, m + l)
+    assert X.rowblocksizes == [n, k]
+    assert X.colblocksizes == [m, l]
+    C = MatrixSymbol('C', n, m)
+    D = MatrixSymbol('D', k, l)
+    Y = BlockDiagMatrix(C, D)
+    assert block_collapse(X + Y) == BlockDiagMatrix(A + C, B + D)
+    assert block_collapse(X * Y.T) == BlockDiagMatrix(A * C.T, B * D.T)
+    raises(NonInvertibleMatrixError, lambda: BlockDiagMatrix(A, C.T).inverse())
+
+def test_BlockDiagMatrix_determinant():
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', m, m)
+    assert det(BlockDiagMatrix()) == 1
+    assert det(BlockDiagMatrix(A)) == det(A)
+    assert det(BlockDiagMatrix(A, B)) == det(A) * det(B)
+
+    # non-square blocks
+    C = MatrixSymbol('C', m, n)
+    D = MatrixSymbol('D', n, m)
+    assert det(BlockDiagMatrix(C, D)) == 0
+
+def test_BlockDiagMatrix_trace():
+    assert trace(BlockDiagMatrix()) == 0
+    assert trace(BlockDiagMatrix(ZeroMatrix(n, n))) == 0
+    A = MatrixSymbol('A', n, n)
+    assert trace(BlockDiagMatrix(A)) == trace(A)
+    B = MatrixSymbol('B', m, m)
+    assert trace(BlockDiagMatrix(A, B)) == trace(A) + trace(B)
+
+    # non-square blocks
+    C = MatrixSymbol('C', m, n)
+    D = MatrixSymbol('D', n, m)
+    assert isinstance(trace(BlockDiagMatrix(C, D)), Trace)
+
+def test_BlockDiagMatrix_transpose():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', k, l)
+    assert transpose(BlockDiagMatrix()) == BlockDiagMatrix()
+    assert transpose(BlockDiagMatrix(A)) == BlockDiagMatrix(A.T)
+    assert transpose(BlockDiagMatrix(A, B)) == BlockDiagMatrix(A.T, B.T)
+
+def test_issue_2460():
+    bdm1 = BlockDiagMatrix(Matrix([i]), Matrix([j]))
+    bdm2 = BlockDiagMatrix(Matrix([k]), Matrix([l]))
+    assert block_collapse(bdm1 + bdm2) == BlockDiagMatrix(Matrix([i + k]), Matrix([j + l]))
+
+def test_blockcut():
+    A = MatrixSymbol('A', n, m)
+    B = blockcut(A, (n/2, n/2), (m/2, m/2))
+    assert B == BlockMatrix([[A[:n/2, :m/2], A[:n/2, m/2:]],
+                             [A[n/2:, :m/2], A[n/2:, m/2:]]])
+
+    M = ImmutableMatrix(4, 4, range(16))
+    B = blockcut(M, (2, 2), (2, 2))
+    assert M == ImmutableMatrix(B)
+
+    B = blockcut(M, (1, 3), (2, 2))
+    assert ImmutableMatrix(B.blocks[0, 1]) == ImmutableMatrix([[2, 3]])
+
+def test_reblock_2x2():
+    B = BlockMatrix([[MatrixSymbol('A_%d%d'%(i,j), 2, 2)
+                            for j in range(3)]
+                            for i in range(3)])
+    assert B.blocks.shape == (3, 3)
+
+    BB = reblock_2x2(B)
+    assert BB.blocks.shape == (2, 2)
+
+    assert B.shape == BB.shape
+    assert B.as_explicit() == BB.as_explicit()
+
+def test_deblock():
+    B = BlockMatrix([[MatrixSymbol('A_%d%d'%(i,j), n, n)
+                    for j in range(4)]
+                    for i in range(4)])
+
+    assert deblock(reblock_2x2(B)) == B
+
+def test_block_collapse_type():
+    bm1 = BlockDiagMatrix(ImmutableMatrix([1]), ImmutableMatrix([2]))
+    bm2 = BlockDiagMatrix(ImmutableMatrix([3]), ImmutableMatrix([4]))
+
+    assert bm1.T.__class__ == BlockDiagMatrix
+    assert block_collapse(bm1 - bm2).__class__ == BlockDiagMatrix
+    assert block_collapse(Inverse(bm1)).__class__ == BlockDiagMatrix
+    assert block_collapse(Transpose(bm1)).__class__ == BlockDiagMatrix
+    assert bc_transpose(Transpose(bm1)).__class__ == BlockDiagMatrix
+    assert bc_inverse(Inverse(bm1)).__class__ == BlockDiagMatrix
+
+def test_invalid_block_matrix():
+    raises(ValueError, lambda: BlockMatrix([
+        [Identity(2), Identity(5)],
+    ]))
+    raises(ValueError, lambda: BlockMatrix([
+        [Identity(n), Identity(m)],
+    ]))
+    raises(ValueError, lambda: BlockMatrix([
+        [ZeroMatrix(n, n), ZeroMatrix(n, n)],
+        [ZeroMatrix(n, n - 1), ZeroMatrix(n, n + 1)],
+    ]))
+    raises(ValueError, lambda: BlockMatrix([
+        [ZeroMatrix(n - 1, n), ZeroMatrix(n, n)],
+        [ZeroMatrix(n + 1, n), ZeroMatrix(n, n)],
+    ]))
+
+def test_block_lu_decomposition():
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', n, m)
+    C = MatrixSymbol('C', m, n)
+    D = MatrixSymbol('D', m, m)
+    X = BlockMatrix([[A, B], [C, D]])
+
+    #LDU decomposition
+    L, D, U = X.LDUdecomposition()
+    assert block_collapse(L*D*U) == X
+
+    #UDL decomposition
+    U, D, L = X.UDLdecomposition()
+    assert block_collapse(U*D*L) == X
+
+    #LU decomposition
+    L, U = X.LUdecomposition()
+    assert block_collapse(L*U) == X
+
+def test_issue_21866():
+    n  = 10
+    I  = Identity(n)
+    O  = ZeroMatrix(n, n)
+    A  = BlockMatrix([[  I,  O,  O,  O ],
+                      [  O,  I,  O,  O ],
+                      [  O,  O,  I,  O ],
+                      [  I,  O,  O,  I ]])
+    Ainv = block_collapse(A.inv())
+    AinvT = BlockMatrix([[  I,  O,  O,  O ],
+                      [  O,  I,  O,  O ],
+                      [  O,  O,  I,  O ],
+                      [  -I,  O,  O,  I ]])
+    assert Ainv == AinvT
+
+
+def test_adjoint_and_special_matrices():
+    A = Identity(3)
+    B = OneMatrix(3, 2)
+    C = ZeroMatrix(2, 3)
+    D = Identity(2)
+    X = BlockMatrix([[A, B], [C, D]])
+    X2 = BlockMatrix([[A, S.ImaginaryUnit*B], [C, D]])
+    assert X.adjoint() == BlockMatrix([[A, ZeroMatrix(3, 2)], [OneMatrix(2, 3), D]])
+    assert re(X) == X
+    assert X2.adjoint() == BlockMatrix([[A, ZeroMatrix(3, 2)], [-S.ImaginaryUnit*OneMatrix(2, 3), D]])
+    assert im(X2) == BlockMatrix([[ZeroMatrix(3, 3), OneMatrix(3, 2)], [ZeroMatrix(2, 3), ZeroMatrix(2, 2)]])

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_companion.py

@@ -0,0 +1,48 @@
+from sympy.core.expr import unchanged
+from sympy.core.symbol import Symbol, symbols
+from sympy.matrices.immutable import ImmutableDenseMatrix
+from sympy.matrices.expressions.companion import CompanionMatrix
+from sympy.polys.polytools import Poly
+from sympy.testing.pytest import raises
+
+
+def test_creation():
+    x = Symbol('x')
+    y = Symbol('y')
+    raises(ValueError, lambda: CompanionMatrix(1))
+    raises(ValueError, lambda: CompanionMatrix(Poly([1], x)))
+    raises(ValueError, lambda: CompanionMatrix(Poly([2, 1], x)))
+    raises(ValueError, lambda: CompanionMatrix(Poly(x*y, [x, y])))
+    assert unchanged(CompanionMatrix, Poly([1, 2, 3], x))
+
+
+def test_shape():
+    c0, c1, c2 = symbols('c0:3')
+    x = Symbol('x')
+    assert CompanionMatrix(Poly([1, c0], x)).shape == (1, 1)
+    assert CompanionMatrix(Poly([1, c1, c0], x)).shape == (2, 2)
+    assert CompanionMatrix(Poly([1, c2, c1, c0], x)).shape == (3, 3)
+
+
+def test_entry():
+    c0, c1, c2 = symbols('c0:3')
+    x = Symbol('x')
+    A = CompanionMatrix(Poly([1, c2, c1, c0], x))
+    assert A[0, 0] == 0
+    assert A[1, 0] == 1
+    assert A[1, 1] == 0
+    assert A[2, 1] == 1
+    assert A[0, 2] == -c0
+    assert A[1, 2] == -c1
+    assert A[2, 2] == -c2
+
+
+def test_as_explicit():
+    c0, c1, c2 = symbols('c0:3')
+    x = Symbol('x')
+    assert CompanionMatrix(Poly([1, c0], x)).as_explicit() == \
+        ImmutableDenseMatrix([-c0])
+    assert CompanionMatrix(Poly([1, c1, c0], x)).as_explicit() == \
+        ImmutableDenseMatrix([[0, -c0], [1, -c1]])
+    assert CompanionMatrix(Poly([1, c2, c1, c0], x)).as_explicit() == \
+        ImmutableDenseMatrix([[0, 0, -c0], [1, 0, -c1], [0, 1, -c2]])

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_derivatives.py

@@ -0,0 +1,477 @@
+"""
+Some examples have been taken from:
+
+http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf
+"""
+from sympy import KroneckerProduct
+from sympy.combinatorics import Permutation
+from sympy.concrete.summations import Sum
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin, tan)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.matrices.expressions.determinant import Determinant
+from sympy.matrices.expressions.diagonal import DiagMatrix
+from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct, hadamard_product)
+from sympy.matrices.expressions.inverse import Inverse
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.special import OneMatrix
+from sympy.matrices.expressions.trace import Trace
+from sympy.matrices.expressions.matadd import MatAdd
+from sympy.matrices.expressions.matmul import MatMul
+from sympy.matrices.expressions.special import (Identity, ZeroMatrix)
+from sympy.tensor.array.array_derivatives import ArrayDerivative
+from sympy.matrices.expressions import hadamard_power
+from sympy.tensor.array.expressions.array_expressions import ArrayAdd, ArrayTensorProduct, PermuteDims
+
+i, j, k = symbols("i j k")
+m, n = symbols("m n")
+
+X = MatrixSymbol("X", k, k)
+x = MatrixSymbol("x", k, 1)
+y = MatrixSymbol("y", k, 1)
+
+A = MatrixSymbol("A", k, k)
+B = MatrixSymbol("B", k, k)
+C = MatrixSymbol("C", k, k)
+D = MatrixSymbol("D", k, k)
+
+a = MatrixSymbol("a", k, 1)
+b = MatrixSymbol("b", k, 1)
+c = MatrixSymbol("c", k, 1)
+d = MatrixSymbol("d", k, 1)
+
+
+KDelta = lambda i, j: KroneckerDelta(i, j, (0, k-1))
+
+
+def _check_derivative_with_explicit_matrix(expr, x, diffexpr, dim=2):
+    # TODO: this is commented because it slows down the tests.
+    return
+
+    expr = expr.xreplace({k: dim})
+    x = x.xreplace({k: dim})
+    diffexpr = diffexpr.xreplace({k: dim})
+
+    expr = expr.as_explicit()
+    x = x.as_explicit()
+    diffexpr = diffexpr.as_explicit()
+
+    assert expr.diff(x).reshape(*diffexpr.shape).tomatrix() == diffexpr
+
+
+def test_matrix_derivative_by_scalar():
+    assert A.diff(i) == ZeroMatrix(k, k)
+    assert (A*(X + B)*c).diff(i) == ZeroMatrix(k, 1)
+    assert x.diff(i) == ZeroMatrix(k, 1)
+    assert (x.T*y).diff(i) == ZeroMatrix(1, 1)
+    assert (x*x.T).diff(i) == ZeroMatrix(k, k)
+    assert (x + y).diff(i) == ZeroMatrix(k, 1)
+    assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1)
+    assert hadamard_power(x, i).diff(i).dummy_eq(
+        HadamardProduct(x.applyfunc(log), HadamardPower(x, i)))
+    assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1)
+    assert hadamard_product(i*OneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y)
+    assert (i*x).diff(i) == x
+    assert (sin(i)*A*B*x).diff(i) == cos(i)*A*B*x
+    assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1)
+    assert Trace(i**2*X).diff(i) == 2*i*Trace(X)
+
+    mu = symbols("mu")
+    expr = (2*mu*x)
+    assert expr.diff(x) == 2*mu*Identity(k)
+
+
+def test_one_matrix():
+    assert MatMul(x.T, OneMatrix(k, 1)).diff(x) == OneMatrix(k, 1)
+
+
+def test_matrix_derivative_non_matrix_result():
+    # This is a 4-dimensional array:
+    I = Identity(k)
+    AdA = PermuteDims(ArrayTensorProduct(I, I), Permutation(3)(1, 2))
+    assert A.diff(A) == AdA
+    assert A.T.diff(A) == PermuteDims(ArrayTensorProduct(I, I), Permutation(3)(1, 2, 3))
+    assert (2*A).diff(A) == PermuteDims(ArrayTensorProduct(2*I, I), Permutation(3)(1, 2))
+    assert MatAdd(A, A).diff(A) == ArrayAdd(AdA, AdA)
+    assert (A + B).diff(A) == AdA
+
+
+def test_matrix_derivative_trivial_cases():
+    # Cookbook example 33:
+    # TODO: find a way to represent a four-dimensional zero-array:
+    assert X.diff(A) == ArrayDerivative(X, A)
+
+
+def test_matrix_derivative_with_inverse():
+
+    # Cookbook example 61:
+    expr = a.T*Inverse(X)*b
+    assert expr.diff(X) == -Inverse(X).T*a*b.T*Inverse(X).T
+
+    # Cookbook example 62:
+    expr = Determinant(Inverse(X))
+    # Not implemented yet:
+    # assert expr.diff(X) == -Determinant(X.inv())*(X.inv()).T
+
+    # Cookbook example 63:
+    expr = Trace(A*Inverse(X)*B)
+    assert expr.diff(X) == -(X**(-1)*B*A*X**(-1)).T
+
+    # Cookbook example 64:
+    expr = Trace(Inverse(X + A))
+    assert expr.diff(X) == -(Inverse(X + A)).T**2
+
+
+def test_matrix_derivative_vectors_and_scalars():
+
+    assert x.diff(x) == Identity(k)
+    assert x[i, 0].diff(x[m, 0]).doit() == KDelta(m, i)
+
+    assert x.T.diff(x) == Identity(k)
+
+    # Cookbook example 69:
+    expr = x.T*a
+    assert expr.diff(x) == a
+    assert expr[0, 0].diff(x[m, 0]).doit() == a[m, 0]
+    expr = a.T*x
+    assert expr.diff(x) == a
+
+    # Cookbook example 70:
+    expr = a.T*X*b
+    assert expr.diff(X) == a*b.T
+
+    # Cookbook example 71:
+    expr = a.T*X.T*b
+    assert expr.diff(X) == b*a.T
+
+    # Cookbook example 72:
+    expr = a.T*X*a
+    assert expr.diff(X) == a*a.T
+    expr = a.T*X.T*a
+    assert expr.diff(X) == a*a.T
+
+    # Cookbook example 77:
+    expr = b.T*X.T*X*c
+    assert expr.diff(X) == X*b*c.T + X*c*b.T
+
+    # Cookbook example 78:
+    expr = (B*x + b).T*C*(D*x + d)
+    assert expr.diff(x) == B.T*C*(D*x + d) + D.T*C.T*(B*x + b)
+
+    # Cookbook example 81:
+    expr = x.T*B*x
+    assert expr.diff(x) == B*x + B.T*x
+
+    # Cookbook example 82:
+    expr = b.T*X.T*D*X*c
+    assert expr.diff(X) == D.T*X*b*c.T + D*X*c*b.T
+
+    # Cookbook example 83:
+    expr = (X*b + c).T*D*(X*b + c)
+    assert expr.diff(X) == D*(X*b + c)*b.T + D.T*(X*b + c)*b.T
+    assert str(expr[0, 0].diff(X[m, n]).doit()) == \
+        'b[n, 0]*Sum((c[_i_1, 0] + Sum(X[_i_1, _i_3]*b[_i_3, 0], (_i_3, 0, k - 1)))*D[_i_1, m], (_i_1, 0, k - 1)) + Sum((c[_i_2, 0] + Sum(X[_i_2, _i_4]*b[_i_4, 0], (_i_4, 0, k - 1)))*D[m, _i_2]*b[n, 0], (_i_2, 0, k - 1))'
+
+    # See https://github.com/sympy/sympy/issues/16504#issuecomment-1018339957
+    expr = x*x.T*x
+    I = Identity(k)
+    assert expr.diff(x) == KroneckerProduct(I, x.T*x) + 2*x*x.T
+
+
+def test_matrix_derivatives_of_traces():
+
+    expr = Trace(A)*A
+    I = Identity(k)
+    assert expr.diff(A) == ArrayAdd(ArrayTensorProduct(I, A), PermuteDims(ArrayTensorProduct(Trace(A)*I, I), Permutation(3)(1, 2)))
+    assert expr[i, j].diff(A[m, n]).doit() == (
+        KDelta(i, m)*KDelta(j, n)*Trace(A) +
+        KDelta(m, n)*A[i, j]
+    )
+
+    ## First order:
+
+    # Cookbook example 99:
+    expr = Trace(X)
+    assert expr.diff(X) == Identity(k)
+    assert expr.rewrite(Sum).diff(X[m, n]).doit() == KDelta(m, n)
+
+    # Cookbook example 100:
+    expr = Trace(X*A)
+    assert expr.diff(X) == A.T
+    assert expr.rewrite(Sum).diff(X[m, n]).doit() == A[n, m]
+
+    # Cookbook example 101:
+    expr = Trace(A*X*B)
+    assert expr.diff(X) == A.T*B.T
+    assert expr.rewrite(Sum).diff(X[m, n]).doit().dummy_eq((A.T*B.T)[m, n])
+
+    # Cookbook example 102:
+    expr = Trace(A*X.T*B)
+    assert expr.diff(X) == B*A
+
+    # Cookbook example 103:
+    expr = Trace(X.T*A)
+    assert expr.diff(X) == A
+
+    # Cookbook example 104:
+    expr = Trace(A*X.T)
+    assert expr.diff(X) == A
+
+    # Cookbook example 105:
+    # TODO: TensorProduct is not supported
+    #expr = Trace(TensorProduct(A, X))
+    #assert expr.diff(X) == Trace(A)*Identity(k)
+
+    ## Second order:
+
+    # Cookbook example 106:
+    expr = Trace(X**2)
+    assert expr.diff(X) == 2*X.T
+
+    # Cookbook example 107:
+    expr = Trace(X**2*B)
+    assert expr.diff(X) == (X*B + B*X).T
+    expr = Trace(MatMul(X, X, B))
+    assert expr.diff(X) == (X*B + B*X).T
+
+    # Cookbook example 108:
+    expr = Trace(X.T*B*X)
+    assert expr.diff(X) == B*X + B.T*X
+
+    # Cookbook example 109:
+    expr = Trace(B*X*X.T)
+    assert expr.diff(X) == B*X + B.T*X
+
+    # Cookbook example 110:
+    expr = Trace(X*X.T*B)
+    assert expr.diff(X) == B*X + B.T*X
+
+    # Cookbook example 111:
+    expr = Trace(X*B*X.T)
+    assert expr.diff(X) == X*B.T + X*B
+
+    # Cookbook example 112:
+    expr = Trace(B*X.T*X)
+    assert expr.diff(X) == X*B.T + X*B
+
+    # Cookbook example 113:
+    expr = Trace(X.T*X*B)
+    assert expr.diff(X) == X*B.T + X*B
+
+    # Cookbook example 114:
+    expr = Trace(A*X*B*X)
+    assert expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T
+
+    # Cookbook example 115:
+    expr = Trace(X.T*X)
+    assert expr.diff(X) == 2*X
+    expr = Trace(X*X.T)
+    assert expr.diff(X) == 2*X
+
+    # Cookbook example 116:
+    expr = Trace(B.T*X.T*C*X*B)
+    assert expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T
+
+    # Cookbook example 117:
+    expr = Trace(X.T*B*X*C)
+    assert expr.diff(X) == B*X*C + B.T*X*C.T
+
+    # Cookbook example 118:
+    expr = Trace(A*X*B*X.T*C)
+    assert expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B
+
+    # Cookbook example 119:
+    expr = Trace((A*X*B + C)*(A*X*B + C).T)
+    assert expr.diff(X) == 2*A.T*(A*X*B + C)*B.T
+
+    # Cookbook example 120:
+    # TODO: no support for TensorProduct.
+    # expr = Trace(TensorProduct(X, X))
+    # expr = Trace(X)*Trace(X)
+    # expr.diff(X) == 2*Trace(X)*Identity(k)
+
+    # Higher Order
+
+    # Cookbook example 121:
+    expr = Trace(X**k)
+    #assert expr.diff(X) == k*(X**(k-1)).T
+
+    # Cookbook example 122:
+    expr = Trace(A*X**k)
+    #assert expr.diff(X) == # Needs indices
+
+    # Cookbook example 123:
+    expr = Trace(B.T*X.T*C*X*X.T*C*X*B)
+    assert expr.diff(X) == C*X*X.T*C*X*B*B.T + C.T*X*B*B.T*X.T*C.T*X + C*X*B*B.T*X.T*C*X + C.T*X*X.T*C.T*X*B*B.T
+
+    # Other
+
+    # Cookbook example 124:
+    expr = Trace(A*X**(-1)*B)
+    assert expr.diff(X) == -Inverse(X).T*A.T*B.T*Inverse(X).T
+
+    # Cookbook example 125:
+    expr = Trace(Inverse(X.T*C*X)*A)
+    # Warning: result in the cookbook is equivalent if B and C are symmetric:
+    assert expr.diff(X) == - X.inv().T*A.T*X.inv()*C.inv().T*X.inv().T - X.inv().T*A*X.inv()*C.inv()*X.inv().T
+
+    # Cookbook example 126:
+    expr = Trace((X.T*C*X).inv()*(X.T*B*X))
+    assert expr.diff(X) == -2*C*X*(X.T*C*X).inv()*X.T*B*X*(X.T*C*X).inv() + 2*B*X*(X.T*C*X).inv()
+
+    # Cookbook example 127:
+    expr = Trace((A + X.T*C*X).inv()*(X.T*B*X))
+    # Warning: result in the cookbook is equivalent if B and C are symmetric:
+    assert expr.diff(X) == B*X*Inverse(A + X.T*C*X) - C*X*Inverse(A + X.T*C*X)*X.T*B*X*Inverse(A + X.T*C*X) - C.T*X*Inverse(A.T + (C*X).T*X)*X.T*B.T*X*Inverse(A.T + (C*X).T*X) + B.T*X*Inverse(A.T + (C*X).T*X)
+
+
+def test_derivatives_of_complicated_matrix_expr():
+    expr = a.T*(A*X*(X.T*B + X*A) + B.T*X.T*(a*b.T*(X*D*X.T + X*(X.T*B + A*X)*D*B - X.T*C.T*A)*B + B*(X*D.T + B*A*X*A.T - 3*X*D))*B + 42*X*B*X.T*A.T*(X + X.T))*b
+    result = (B*(B*A*X*A.T - 3*X*D + X*D.T) + a*b.T*(X*(A*X + X.T*B)*D*B + X*D*X.T - X.T*C.T*A)*B)*B*b*a.T*B.T + B**2*b*a.T*B.T*X.T*a*b.T*X*D + 42*A*X*B.T*X.T*a*b.T + B*D*B**3*b*a.T*B.T*X.T*a*b.T*X + B*b*a.T*A*X + a*b.T*(42*X + 42*X.T)*A*X*B.T + b*a.T*X*B*a*b.T*B.T**2*X*D.T + b*a.T*X*B*a*b.T*B.T**3*D.T*(B.T*X + X.T*A.T) + 42*b*a.T*X*B*X.T*A.T + A.T*(42*X + 42*X.T)*b*a.T*X*B + A.T*B.T**2*X*B*a*b.T*B.T*A + A.T*a*b.T*(A.T*X.T + B.T*X) + A.T*X.T*b*a.T*X*B*a*b.T*B.T**3*D.T + B.T*X*B*a*b.T*B.T*D - 3*B.T*X*B*a*b.T*B.T*D.T - C.T*A*B**2*b*a.T*B.T*X.T*a*b.T + X.T*A.T*a*b.T*A.T
+    assert expr.diff(X) == result
+
+
+def test_mixed_deriv_mixed_expressions():
+
+    expr = 3*Trace(A)
+    assert expr.diff(A) == 3*Identity(k)
+
+    expr = k
+    deriv = expr.diff(A)
+    assert isinstance(deriv, ZeroMatrix)
+    assert deriv == ZeroMatrix(k, k)
+
+    expr = Trace(A)**2
+    assert expr.diff(A) == (2*Trace(A))*Identity(k)
+
+    expr = Trace(A)*A
+    I = Identity(k)
+    assert expr.diff(A) == ArrayAdd(ArrayTensorProduct(I, A), PermuteDims(ArrayTensorProduct(Trace(A)*I, I), Permutation(3)(1, 2)))
+
+    expr = Trace(Trace(A)*A)
+    assert expr.diff(A) == (2*Trace(A))*Identity(k)
+
+    expr = Trace(Trace(Trace(A)*A)*A)
+    assert expr.diff(A) == (3*Trace(A)**2)*Identity(k)
+
+
+def test_derivatives_matrix_norms():
+
+    expr = x.T*y
+    assert expr.diff(x) == y
+    assert expr[0, 0].diff(x[m, 0]).doit() == y[m, 0]
+
+    expr = (x.T*y)**S.Half
+    assert expr.diff(x) == y/(2*sqrt(x.T*y))
+
+    expr = (x.T*x)**S.Half
+    assert expr.diff(x) == x*(x.T*x)**Rational(-1, 2)
+
+    expr = (c.T*a*x.T*b)**S.Half
+    assert expr.diff(x) == b*a.T*c/sqrt(c.T*a*x.T*b)/2
+
+    expr = (c.T*a*x.T*b)**Rational(1, 3)
+    assert expr.diff(x) == b*a.T*c*(c.T*a*x.T*b)**Rational(-2, 3)/3
+
+    expr = (a.T*X*b)**S.Half
+    assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*b.T
+
+    expr = d.T*x*(a.T*X*b)**S.Half*y.T*c
+    assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*x.T*d*y.T*c*b.T
+
+
+def test_derivatives_elementwise_applyfunc():
+
+    expr = x.applyfunc(tan)
+    assert expr.diff(x).dummy_eq(
+        DiagMatrix(x.applyfunc(lambda x: tan(x)**2 + 1)))
+    assert expr[i, 0].diff(x[m, 0]).doit() == (tan(x[i, 0])**2 + 1)*KDelta(i, m)
+    _check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
+
+    expr = (i**2*x).applyfunc(sin)
+    assert expr.diff(i).dummy_eq(
+        HadamardProduct((2*i)*x, (i**2*x).applyfunc(cos)))
+    assert expr[i, 0].diff(i).doit() == 2*i*x[i, 0]*cos(i**2*x[i, 0])
+    _check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
+
+    expr = (log(i)*A*B).applyfunc(sin)
+    assert expr.diff(i).dummy_eq(
+        HadamardProduct(A*B/i, (log(i)*A*B).applyfunc(cos)))
+    _check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
+
+    expr = A*x.applyfunc(exp)
+    # TODO: restore this result (currently returning the transpose):
+    #  assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(exp))*A.T)
+    _check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
+
+    expr = x.T*A*x + k*y.applyfunc(sin).T*x
+    assert expr.diff(x).dummy_eq(A.T*x + A*x + k*y.applyfunc(sin))
+    _check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
+
+    expr = x.applyfunc(sin).T*y
+    # TODO: restore (currently returning the transpose):
+    #  assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(cos))*y)
+    _check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
+
+    expr = (a.T * X * b).applyfunc(sin)
+    assert expr.diff(X).dummy_eq(a*(a.T*X*b).applyfunc(cos)*b.T)
+    _check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
+
+    expr = a.T * X.applyfunc(sin) * b
+    assert expr.diff(X).dummy_eq(
+        DiagMatrix(a)*X.applyfunc(cos)*DiagMatrix(b))
+    _check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
+
+    expr = a.T * (A*X*B).applyfunc(sin) * b
+    assert expr.diff(X).dummy_eq(
+        A.T*DiagMatrix(a)*(A*X*B).applyfunc(cos)*DiagMatrix(b)*B.T)
+    _check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
+
+    expr = a.T * (A*X*b).applyfunc(sin) * b.T
+    # TODO: not implemented
+    #assert expr.diff(X) == ...
+    #_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
+
+    expr = a.T*A*X.applyfunc(sin)*B*b
+    assert expr.diff(X).dummy_eq(
+        HadamardProduct(A.T * a * b.T * B.T, X.applyfunc(cos)))
+
+    expr = a.T * (A*X.applyfunc(sin)*B).applyfunc(log) * b
+    # TODO: wrong
+    # assert expr.diff(X) == A.T*DiagMatrix(a)*(A*X.applyfunc(sin)*B).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)*B.T
+
+    expr = a.T * (X.applyfunc(sin)).applyfunc(log) * b
+    # TODO: wrong
+    # assert expr.diff(X) == DiagMatrix(a)*X.applyfunc(sin).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)
+
+
+def test_derivatives_of_hadamard_expressions():
+
+    # Hadamard Product
+
+    expr = hadamard_product(a, x, b)
+    assert expr.diff(x) == DiagMatrix(hadamard_product(b, a))
+
+    expr = a.T*hadamard_product(A, X, B)*b
+    assert expr.diff(X) == HadamardProduct(a*b.T, A, B)
+
+    # Hadamard Power
+
+    expr = hadamard_power(x, 2)
+    assert expr.diff(x).doit() == 2*DiagMatrix(x)
+
+    expr = hadamard_power(x.T, 2)
+    assert expr.diff(x).doit() == 2*DiagMatrix(x)
+
+    expr = hadamard_power(x, S.Half)
+    assert expr.diff(x) == S.Half*DiagMatrix(hadamard_power(x, Rational(-1, 2)))
+
+    expr = hadamard_power(a.T*X*b, 2)
+    assert expr.diff(X) == 2*a*a.T*X*b*b.T
+
+    expr = hadamard_power(a.T*X*b, S.Half)
+    assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*b.T

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_determinant.py

@@ -0,0 +1,62 @@
+from sympy.core import S, symbols
+from sympy.matrices import eye, ones, Matrix, ShapeError
+from sympy.matrices.expressions import (
+    Identity, MatrixExpr, MatrixSymbol, Determinant,
+    det, per, ZeroMatrix, Transpose,
+    Permanent
+)
+from sympy.matrices.expressions.special import OneMatrix
+from sympy.testing.pytest import raises
+from sympy.assumptions.ask import Q
+from sympy.assumptions.refine import refine
+
+n = symbols('n', integer=True)
+A = MatrixSymbol('A', n, n)
+B = MatrixSymbol('B', n, n)
+C = MatrixSymbol('C', 3, 4)
+
+
+def test_det():
+    assert isinstance(Determinant(A), Determinant)
+    assert not isinstance(Determinant(A), MatrixExpr)
+    raises(ShapeError, lambda: Determinant(C))
+    assert det(eye(3)) == 1
+    assert det(Matrix(3, 3, [1, 3, 2, 4, 1, 3, 2, 5, 2])) == 17
+    _ = A / det(A)  # Make sure this is possible
+
+    raises(TypeError, lambda: Determinant(S.One))
+
+    assert Determinant(A).arg is A
+
+def test_eval_determinant():
+    assert det(Identity(n)) == 1
+    assert det(ZeroMatrix(n, n)) == 0
+    assert det(OneMatrix(n, n)) == Determinant(OneMatrix(n, n))
+    assert det(OneMatrix(1, 1)) == 1
+    assert det(OneMatrix(2, 2)) == 0
+    assert det(Transpose(A)) == det(A)
+
+
+def test_refine():
+    assert refine(det(A), Q.orthogonal(A)) == 1
+    assert refine(det(A), Q.singular(A)) == 0
+    assert refine(det(A), Q.unit_triangular(A)) == 1
+    assert refine(det(A), Q.normal(A)) == det(A)
+
+
+def test_commutative():
+    det_a = Determinant(A)
+    det_b = Determinant(B)
+    assert det_a.is_commutative
+    assert det_b.is_commutative
+    assert det_a * det_b == det_b * det_a
+
+def test_permanent():
+    assert isinstance(Permanent(A), Permanent)
+    assert not isinstance(Permanent(A), MatrixExpr)
+    assert isinstance(Permanent(C), Permanent)
+    assert Permanent(ones(3, 3)).doit() == 6
+    _ = C / per(C)
+    assert per(Matrix(3, 3, [1, 3, 2, 4, 1, 3, 2, 5, 2])) == 103
+    raises(TypeError, lambda: Permanent(S.One))
+    assert Permanent(A).arg is A

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_diagonal.py

@@ -0,0 +1,156 @@
+from sympy.matrices.expressions import MatrixSymbol
+from sympy.matrices.expressions.diagonal import DiagonalMatrix, DiagonalOf, DiagMatrix, diagonalize_vector
+from sympy.assumptions.ask import (Q, ask)
+from sympy.core.symbol import Symbol
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.matmul import MatMul
+from sympy.matrices.expressions.special import Identity
+from sympy.testing.pytest import raises
+
+
+n = Symbol('n')
+m = Symbol('m')
+
+
+def test_DiagonalMatrix():
+    x = MatrixSymbol('x', n, m)
+    D = DiagonalMatrix(x)
+    assert D.diagonal_length is None
+    assert D.shape == (n, m)
+
+    x = MatrixSymbol('x', n, n)
+    D = DiagonalMatrix(x)
+    assert D.diagonal_length == n
+    assert D.shape == (n, n)
+    assert D[1, 2] == 0
+    assert D[1, 1] == x[1, 1]
+    i = Symbol('i')
+    j = Symbol('j')
+    x = MatrixSymbol('x', 3, 3)
+    ij = DiagonalMatrix(x)[i, j]
+    assert ij != 0
+    assert ij.subs({i:0, j:0}) == x[0, 0]
+    assert ij.subs({i:0, j:1}) == 0
+    assert ij.subs({i:1, j:1}) == x[1, 1]
+    assert ask(Q.diagonal(D))  # affirm that D is diagonal
+
+    x = MatrixSymbol('x', n, 3)
+    D = DiagonalMatrix(x)
+    assert D.diagonal_length == 3
+    assert D.shape == (n, 3)
+    assert D[2, m] == KroneckerDelta(2, m)*x[2, m]
+    assert D[3, m] == 0
+    raises(IndexError, lambda: D[m, 3])
+
+    x = MatrixSymbol('x', 3, n)
+    D = DiagonalMatrix(x)
+    assert D.diagonal_length == 3
+    assert D.shape == (3, n)
+    assert D[m, 2] == KroneckerDelta(m, 2)*x[m, 2]
+    assert D[m, 3] == 0
+    raises(IndexError, lambda: D[3, m])
+
+    x = MatrixSymbol('x', n, m)
+    D = DiagonalMatrix(x)
+    assert D.diagonal_length is None
+    assert D.shape == (n, m)
+    assert D[m, 4] != 0
+
+    x = MatrixSymbol('x', 3, 4)
+    assert [DiagonalMatrix(x)[i] for i in range(12)] == [
+        x[0, 0], 0, 0, 0, 0, x[1, 1], 0, 0, 0, 0, x[2, 2], 0]
+
+    # shape is retained, issue 12427
+    assert (
+        DiagonalMatrix(MatrixSymbol('x', 3, 4))*
+        DiagonalMatrix(MatrixSymbol('x', 4, 2))).shape == (3, 2)
+
+
+def test_DiagonalOf():
+    x = MatrixSymbol('x', n, n)
+    d = DiagonalOf(x)
+    assert d.shape == (n, 1)
+    assert d.diagonal_length == n
+    assert d[2, 0] == d[2] == x[2, 2]
+
+    x = MatrixSymbol('x', n, m)
+    d = DiagonalOf(x)
+    assert d.shape == (None, 1)
+    assert d.diagonal_length is None
+    assert d[2, 0] == d[2] == x[2, 2]
+
+    d = DiagonalOf(MatrixSymbol('x', 4, 3))
+    assert d.shape == (3, 1)
+    d = DiagonalOf(MatrixSymbol('x', n, 3))
+    assert d.shape == (3, 1)
+    d = DiagonalOf(MatrixSymbol('x', 3, n))
+    assert d.shape == (3, 1)
+    x = MatrixSymbol('x', n, m)
+    assert [DiagonalOf(x)[i] for i in range(4)] ==[
+        x[0, 0], x[1, 1], x[2, 2], x[3, 3]]
+
+
+def test_DiagMatrix():
+    x = MatrixSymbol('x', n, 1)
+    d = DiagMatrix(x)
+    assert d.shape == (n, n)
+    assert d[0, 1] == 0
+    assert d[0, 0] == x[0, 0]
+
+    a = MatrixSymbol('a', 1, 1)
+    d = diagonalize_vector(a)
+    assert isinstance(d, MatrixSymbol)
+    assert a == d
+    assert diagonalize_vector(Identity(3)) == Identity(3)
+    assert DiagMatrix(Identity(3)).doit() == Identity(3)
+    assert isinstance(DiagMatrix(Identity(3)), DiagMatrix)
+
+    # A diagonal matrix is equal to its transpose:
+    assert DiagMatrix(x).T == DiagMatrix(x)
+    assert diagonalize_vector(x.T) == DiagMatrix(x)
+
+    dx = DiagMatrix(x)
+    assert dx[0, 0] == x[0, 0]
+    assert dx[1, 1] == x[1, 0]
+    assert dx[0, 1] == 0
+    assert dx[0, m] == x[0, 0]*KroneckerDelta(0, m)
+
+    z = MatrixSymbol('z', 1, n)
+    dz = DiagMatrix(z)
+    assert dz[0, 0] == z[0, 0]
+    assert dz[1, 1] == z[0, 1]
+    assert dz[0, 1] == 0
+    assert dz[0, m] == z[0, m]*KroneckerDelta(0, m)
+
+    v = MatrixSymbol('v', 3, 1)
+    dv = DiagMatrix(v)
+    assert dv.as_explicit() == Matrix([
+        [v[0, 0], 0, 0],
+        [0, v[1, 0], 0],
+        [0, 0, v[2, 0]],
+    ])
+
+    v = MatrixSymbol('v', 1, 3)
+    dv = DiagMatrix(v)
+    assert dv.as_explicit() == Matrix([
+        [v[0, 0], 0, 0],
+        [0, v[0, 1], 0],
+        [0, 0, v[0, 2]],
+    ])
+
+    dv = DiagMatrix(3*v)
+    assert dv.args == (3*v,)
+    assert dv.doit() == 3*DiagMatrix(v)
+    assert isinstance(dv.doit(), MatMul)
+
+    a = MatrixSymbol("a", 3, 1).as_explicit()
+    expr = DiagMatrix(a)
+    result = Matrix([
+        [a[0, 0], 0, 0],
+        [0, a[1, 0], 0],
+        [0, 0, a[2, 0]],
+    ])
+    assert expr.doit() == result
+    expr = DiagMatrix(a.T)
+    assert expr.doit() == result

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_dotproduct.py

@@ -0,0 +1,35 @@
+from sympy.core.expr import unchanged
+from sympy.core.mul import Mul
+from sympy.matrices import Matrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.dotproduct import DotProduct
+from sympy.testing.pytest import raises
+
+
+A = Matrix(3, 1, [1, 2, 3])
+B = Matrix(3, 1, [1, 3, 5])
+C = Matrix(4, 1, [1, 2, 4, 5])
+D = Matrix(2, 2, [1, 2, 3, 4])
+
+def test_docproduct():
+    assert DotProduct(A, B).doit() == 22
+    assert DotProduct(A.T, B).doit() == 22
+    assert DotProduct(A, B.T).doit() == 22
+    assert DotProduct(A.T, B.T).doit() == 22
+
+    raises(TypeError, lambda: DotProduct(1, A))
+    raises(TypeError, lambda: DotProduct(A, 1))
+    raises(TypeError, lambda: DotProduct(A, D))
+    raises(TypeError, lambda: DotProduct(D, A))
+
+    raises(TypeError, lambda: DotProduct(B, C).doit())
+
+def test_dotproduct_symbolic():
+    A = MatrixSymbol('A', 3, 1)
+    B = MatrixSymbol('B', 3, 1)
+
+    dot = DotProduct(A, B)
+    assert dot.is_scalar == True
+    assert unchanged(Mul, 2, dot)
+    # XXX Fix forced evaluation for arithmetics with matrix expressions
+    assert dot * A == (A[0, 0]*B[0, 0] + A[1, 0]*B[1, 0] + A[2, 0]*B[2, 0])*A

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_factorizations.py

@@ -0,0 +1,29 @@
+from sympy.matrices.expressions.factorizations import lu, LofCholesky, qr, svd
+from sympy.assumptions.ask import (Q, ask)
+from sympy.core.symbol import Symbol
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+
+n = Symbol('n')
+X = MatrixSymbol('X', n, n)
+
+def test_LU():
+    L, U = lu(X)
+    assert L.shape == U.shape == X.shape
+    assert ask(Q.lower_triangular(L))
+    assert ask(Q.upper_triangular(U))
+
+def test_Cholesky():
+    LofCholesky(X)
+
+def test_QR():
+    Q_, R = qr(X)
+    assert Q_.shape == R.shape == X.shape
+    assert ask(Q.orthogonal(Q_))
+    assert ask(Q.upper_triangular(R))
+
+def test_svd():
+    U, S, V = svd(X)
+    assert U.shape == S.shape == V.shape == X.shape
+    assert ask(Q.orthogonal(U))
+    assert ask(Q.orthogonal(V))
+    assert ask(Q.diagonal(S))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_fourier.py

@@ -0,0 +1,44 @@
+from sympy.assumptions.ask import (Q, ask)
+from sympy.core.numbers import (I, Rational)
+from sympy.core.singleton import S
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.simplify.simplify import simplify
+from sympy.core.symbol import symbols
+from sympy.matrices.expressions.fourier import DFT, IDFT
+from sympy.matrices import det, Matrix, Identity
+from sympy.testing.pytest import raises
+
+
+def test_dft_creation():
+    assert DFT(2)
+    assert DFT(0)
+    raises(ValueError, lambda: DFT(-1))
+    raises(ValueError, lambda: DFT(2.0))
+    raises(ValueError, lambda: DFT(2 + 1j))
+
+    n = symbols('n')
+    assert DFT(n)
+    n = symbols('n', integer=False)
+    raises(ValueError, lambda: DFT(n))
+    n = symbols('n', negative=True)
+    raises(ValueError, lambda: DFT(n))
+
+
+def test_dft():
+    n, i, j = symbols('n i j')
+    assert DFT(4).shape == (4, 4)
+    assert ask(Q.unitary(DFT(4)))
+    assert Abs(simplify(det(Matrix(DFT(4))))) == 1
+    assert DFT(n)*IDFT(n) == Identity(n)
+    assert DFT(n)[i, j] == exp(-2*S.Pi*I/n)**(i*j) / sqrt(n)
+
+
+def test_dft2():
+    assert DFT(1).as_explicit() == Matrix([[1]])
+    assert DFT(2).as_explicit() == 1/sqrt(2)*Matrix([[1,1],[1,-1]])
+    assert DFT(4).as_explicit() == Matrix([[S.Half,  S.Half,  S.Half, S.Half],
+                                           [S.Half, -I/2, Rational(-1,2),  I/2],
+                                           [S.Half, Rational(-1,2),  S.Half, Rational(-1,2)],
+                                           [S.Half,  I/2, Rational(-1,2), -I/2]])

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_funcmatrix.py

@@ -0,0 +1,56 @@
+from sympy.core import symbols, Lambda
+from sympy.functions import KroneckerDelta
+from sympy.matrices import Matrix
+from sympy.matrices.expressions import FunctionMatrix, MatrixExpr, Identity
+from sympy.testing.pytest import raises, warns
+from sympy.utilities.exceptions import SymPyDeprecationWarning
+
+
+def test_funcmatrix_creation():
+    i, j, k = symbols('i j k')
+    assert FunctionMatrix(2, 2, Lambda((i, j), 0))
+    assert FunctionMatrix(0, 0, Lambda((i, j), 0))
+
+    raises(ValueError, lambda: FunctionMatrix(-1, 0, Lambda((i, j), 0)))
+    raises(ValueError, lambda: FunctionMatrix(2.0, 0, Lambda((i, j), 0)))
+    raises(ValueError, lambda: FunctionMatrix(2j, 0, Lambda((i, j), 0)))
+    raises(ValueError, lambda: FunctionMatrix(0, -1, Lambda((i, j), 0)))
+    raises(ValueError, lambda: FunctionMatrix(0, 2.0, Lambda((i, j), 0)))
+    raises(ValueError, lambda: FunctionMatrix(0, 2j, Lambda((i, j), 0)))
+
+    raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda(i, 0)))
+    with warns(SymPyDeprecationWarning, test_stacklevel=False):
+        # This raises a deprecation warning from sympify()
+        raises(ValueError, lambda: FunctionMatrix(2, 2, lambda i, j: 0))
+    raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i,), 0)))
+    raises(ValueError, lambda: FunctionMatrix(2, 2, Lambda((i, j, k), 0)))
+    raises(ValueError, lambda: FunctionMatrix(2, 2, i+j))
+    assert FunctionMatrix(2, 2, "lambda i, j: 0") == \
+        FunctionMatrix(2, 2, Lambda((i, j), 0))
+
+    m = FunctionMatrix(2, 2, KroneckerDelta)
+    assert m.as_explicit() == Identity(2).as_explicit()
+    assert m.args[2].dummy_eq(Lambda((i, j), KroneckerDelta(i, j)))
+
+    n = symbols('n')
+    assert FunctionMatrix(n, n, Lambda((i, j), 0))
+    n = symbols('n', integer=False)
+    raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
+    n = symbols('n', negative=True)
+    raises(ValueError, lambda: FunctionMatrix(n, n, Lambda((i, j), 0)))
+
+
+def test_funcmatrix():
+    i, j = symbols('i,j')
+    X = FunctionMatrix(3, 3, Lambda((i, j), i - j))
+    assert X[1, 1] == 0
+    assert X[1, 2] == -1
+    assert X.shape == (3, 3)
+    assert X.rows == X.cols == 3
+    assert Matrix(X) == Matrix(3, 3, lambda i, j: i - j)
+    assert isinstance(X*X + X, MatrixExpr)
+
+
+def test_replace_issue():
+    X = FunctionMatrix(3, 3, KroneckerDelta)
+    assert X.replace(lambda x: True, lambda x: x) == X

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_hadamard.py

@@ -0,0 +1,141 @@
+from sympy.matrices.dense import Matrix, eye
+from sympy.matrices.common import ShapeError
+from sympy.matrices.expressions.matadd import MatAdd
+from sympy.matrices.expressions.special import Identity, OneMatrix, ZeroMatrix
+from sympy.core import symbols
+from sympy.testing.pytest import raises, warns_deprecated_sympy
+
+from sympy.matrices import MatrixSymbol
+from sympy.matrices.expressions import (HadamardProduct, hadamard_product, HadamardPower, hadamard_power)
+
+n, m, k = symbols('n,m,k')
+Z = MatrixSymbol('Z', n, n)
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', n, m)
+C = MatrixSymbol('C', m, k)
+
+
+def test_HadamardProduct():
+    assert HadamardProduct(A, B, A).shape == A.shape
+
+    raises(TypeError,  lambda: HadamardProduct(A, n))
+    raises(TypeError,  lambda: HadamardProduct(A, 1))
+
+    assert HadamardProduct(A, 2*B, -A)[1, 1] == \
+            -2 * A[1, 1] * B[1, 1] * A[1, 1]
+
+    mix = HadamardProduct(Z*A, B)*C
+    assert mix.shape == (n, k)
+
+    assert set(HadamardProduct(A, B, A).T.args) == {A.T, A.T, B.T}
+
+
+def test_HadamardProduct_isnt_commutative():
+    assert HadamardProduct(A, B) != HadamardProduct(B, A)
+
+
+def test_mixed_indexing():
+    X = MatrixSymbol('X', 2, 2)
+    Y = MatrixSymbol('Y', 2, 2)
+    Z = MatrixSymbol('Z', 2, 2)
+
+    assert (X*HadamardProduct(Y, Z))[0, 0] == \
+            X[0, 0]*Y[0, 0]*Z[0, 0] + X[0, 1]*Y[1, 0]*Z[1, 0]
+
+
+def test_canonicalize():
+    X = MatrixSymbol('X', 2, 2)
+    Y = MatrixSymbol('Y', 2, 2)
+    with warns_deprecated_sympy():
+        expr = HadamardProduct(X, check=False)
+    assert isinstance(expr, HadamardProduct)
+    expr2 = expr.doit() # unpack is called
+    assert isinstance(expr2, MatrixSymbol)
+    Z = ZeroMatrix(2, 2)
+    U = OneMatrix(2, 2)
+    assert HadamardProduct(Z, X).doit() == Z
+    assert HadamardProduct(U, X, X, U).doit() == HadamardPower(X, 2)
+    assert HadamardProduct(X, U, Y).doit() == HadamardProduct(X, Y)
+    assert HadamardProduct(X, Z, U, Y).doit() == Z
+
+
+def test_hadamard():
+    m, n, p = symbols('m, n, p', integer=True)
+    A = MatrixSymbol('A', m, n)
+    B = MatrixSymbol('B', m, n)
+    X = MatrixSymbol('X', m, m)
+    I = Identity(m)
+
+    raises(TypeError, lambda: hadamard_product())
+    assert hadamard_product(A) == A
+    assert isinstance(hadamard_product(A, B), HadamardProduct)
+    assert hadamard_product(A, B).doit() == hadamard_product(A, B)
+    assert hadamard_product(X, I) == HadamardProduct(I, X)
+    assert isinstance(hadamard_product(X, I), HadamardProduct)
+
+    a = MatrixSymbol("a", k, 1)
+    expr = MatAdd(ZeroMatrix(k, 1), OneMatrix(k, 1))
+    expr = HadamardProduct(expr, a)
+    assert expr.doit() == a
+
+    raises(ValueError, lambda: HadamardProduct())
+
+
+def test_hadamard_product_with_explicit_mat():
+    A = MatrixSymbol("A", 3, 3).as_explicit()
+    B = MatrixSymbol("B", 3, 3).as_explicit()
+    X = MatrixSymbol("X", 3, 3)
+    expr = hadamard_product(A, B)
+    ret = Matrix([i*j for i, j in zip(A, B)]).reshape(3, 3)
+    assert expr == ret
+    expr = hadamard_product(A, X, B)
+    assert expr == HadamardProduct(ret, X)
+    expr = hadamard_product(eye(3), A)
+    assert expr == Matrix([[A[0, 0], 0, 0], [0, A[1, 1], 0], [0, 0, A[2, 2]]])
+    expr = hadamard_product(eye(3), eye(3))
+    assert expr == eye(3)
+
+
+def test_hadamard_power():
+    m, n, p = symbols('m, n, p', integer=True)
+    A = MatrixSymbol('A', m, n)
+
+    assert hadamard_power(A, 1) == A
+    assert isinstance(hadamard_power(A, 2), HadamardPower)
+    assert hadamard_power(A, n).T == hadamard_power(A.T, n)
+    assert hadamard_power(A, n)[0, 0] == A[0, 0]**n
+    assert hadamard_power(m, n) == m**n
+    raises(ValueError, lambda: hadamard_power(A, A))
+
+
+def test_hadamard_power_explicit():
+    A = MatrixSymbol('A', 2, 2)
+    B = MatrixSymbol('B', 2, 2)
+    a, b = symbols('a b')
+
+    assert HadamardPower(a, b) == a**b
+
+    assert HadamardPower(a, B).as_explicit() == \
+        Matrix([
+            [a**B[0, 0], a**B[0, 1]],
+            [a**B[1, 0], a**B[1, 1]]])
+
+    assert HadamardPower(A, b).as_explicit() == \
+        Matrix([
+            [A[0, 0]**b, A[0, 1]**b],
+            [A[1, 0]**b, A[1, 1]**b]])
+
+    assert HadamardPower(A, B).as_explicit() == \
+        Matrix([
+            [A[0, 0]**B[0, 0], A[0, 1]**B[0, 1]],
+            [A[1, 0]**B[1, 0], A[1, 1]**B[1, 1]]])
+
+
+def test_shape_error():
+    A = MatrixSymbol('A', 2, 3)
+    B = MatrixSymbol('B', 3, 3)
+    raises(ShapeError, lambda: HadamardProduct(A, B))
+    raises(ShapeError, lambda: HadamardPower(A, B))
+    A = MatrixSymbol('A', 3, 2)
+    raises(ShapeError, lambda: HadamardProduct(A, B))
+    raises(ShapeError, lambda: HadamardPower(A, B))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_indexing.py

@@ -0,0 +1,299 @@
+from sympy.concrete.summations import Sum
+from sympy.core.symbol import symbols, Symbol, Dummy
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.matrices.dense import eye
+from sympy.matrices.expressions.blockmatrix import BlockMatrix
+from sympy.matrices.expressions.hadamard import HadamardPower
+from sympy.matrices.expressions.matexpr import (MatrixSymbol,
+    MatrixExpr, MatrixElement)
+from sympy.matrices.expressions.matpow import MatPow
+from sympy.matrices.expressions.special import (ZeroMatrix, Identity,
+    OneMatrix)
+from sympy.matrices.expressions.trace import Trace, trace
+from sympy.matrices.immutable import ImmutableMatrix
+from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
+from sympy.testing.pytest import XFAIL, raises
+
+k, l, m, n = symbols('k l m n', integer=True)
+i, j = symbols('i j', integer=True)
+
+W = MatrixSymbol('W', k, l)
+X = MatrixSymbol('X', l, m)
+Y = MatrixSymbol('Y', l, m)
+Z = MatrixSymbol('Z', m, n)
+
+X1 = MatrixSymbol('X1', m, m)
+X2 = MatrixSymbol('X2', m, m)
+X3 = MatrixSymbol('X3', m, m)
+X4 = MatrixSymbol('X4', m, m)
+
+A = MatrixSymbol('A', 2, 2)
+B = MatrixSymbol('B', 2, 2)
+x = MatrixSymbol('x', 1, 2)
+y = MatrixSymbol('x', 2, 1)
+
+
+def test_symbolic_indexing():
+    x12 = X[1, 2]
+    assert all(s in str(x12) for s in ['1', '2', X.name])
+    # We don't care about the exact form of this.  We do want to make sure
+    # that all of these features are present
+
+
+def test_add_index():
+    assert (X + Y)[i, j] == X[i, j] + Y[i, j]
+
+
+def test_mul_index():
+    assert (A*y)[0, 0] == A[0, 0]*y[0, 0] + A[0, 1]*y[1, 0]
+    assert (A*B).as_mutable() == (A.as_mutable() * B.as_mutable())
+    X = MatrixSymbol('X', n, m)
+    Y = MatrixSymbol('Y', m, k)
+
+    result = (X*Y)[4,2]
+    expected = Sum(X[4, i]*Y[i, 2], (i, 0, m - 1))
+    assert result.args[0].dummy_eq(expected.args[0], i)
+    assert result.args[1][1:] == expected.args[1][1:]
+
+
+def test_pow_index():
+    Q = MatPow(A, 2)
+    assert Q[0, 0] == A[0, 0]**2 + A[0, 1]*A[1, 0]
+    n = symbols("n")
+    Q2 = A**n
+    assert Q2[0, 0] == 2*(
+            -sqrt((A[0, 0] + A[1, 1])**2 - 4*A[0, 0]*A[1, 1] +
+            4*A[0, 1]*A[1, 0])/2 + A[0, 0]/2 + A[1, 1]/2
+        )**n * \
+        A[0, 1]*A[1, 0]/(
+            (sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] +
+                  A[1, 1]**2) + A[0, 0] - A[1, 1])*
+            sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)
+        ) - 2*(
+            sqrt((A[0, 0] + A[1, 1])**2 - 4*A[0, 0]*A[1, 1] +
+            4*A[0, 1]*A[1, 0])/2 + A[0, 0]/2 + A[1, 1]/2
+        )**n * A[0, 1]*A[1, 0]/(
+            (-sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] +
+            A[1, 1]**2) + A[0, 0] - A[1, 1])*
+            sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)
+        )
+
+
+def test_transpose_index():
+    assert X.T[i, j] == X[j, i]
+
+
+def test_Identity_index():
+    I = Identity(3)
+    assert I[0, 0] == I[1, 1] == I[2, 2] == 1
+    assert I[1, 0] == I[0, 1] == I[2, 1] == 0
+    assert I[i, 0].delta_range == (0, 2)
+    raises(IndexError, lambda: I[3, 3])
+
+
+def test_block_index():
+    I = Identity(3)
+    Z = ZeroMatrix(3, 3)
+    B = BlockMatrix([[I, I], [I, I]])
+    e3 = ImmutableMatrix(eye(3))
+    BB = BlockMatrix([[e3, e3], [e3, e3]])
+    assert B[0, 0] == B[3, 0] == B[0, 3] == B[3, 3] == 1
+    assert B[4, 3] == B[5, 1] == 0
+
+    BB = BlockMatrix([[e3, e3], [e3, e3]])
+    assert B.as_explicit() == BB.as_explicit()
+
+    BI = BlockMatrix([[I, Z], [Z, I]])
+
+    assert BI.as_explicit().equals(eye(6))
+
+
+def test_block_index_symbolic():
+    # Note that these matrices may be zero-sized and indices may be negative, which causes
+    # all naive simplifications given in the comments to be invalid
+    A1 = MatrixSymbol('A1', n, k)
+    A2 = MatrixSymbol('A2', n, l)
+    A3 = MatrixSymbol('A3', m, k)
+    A4 = MatrixSymbol('A4', m, l)
+    A = BlockMatrix([[A1, A2], [A3, A4]])
+    assert A[0, 0] == MatrixElement(A, 0, 0)  # Cannot be A1[0, 0]
+    assert A[n - 1, k - 1] == A1[n - 1, k - 1]
+    assert A[n, k] == A4[0, 0]
+    assert A[n + m - 1, 0] == MatrixElement(A, n + m - 1, 0)  # Cannot be A3[m - 1, 0]
+    assert A[0, k + l - 1] == MatrixElement(A, 0, k + l - 1)  # Cannot be A2[0, l - 1]
+    assert A[n + m - 1, k + l - 1] == MatrixElement(A, n + m - 1, k + l - 1)  # Cannot be A4[m - 1, l - 1]
+    assert A[i, j] == MatrixElement(A, i, j)
+    assert A[n + i, k + j] == MatrixElement(A, n + i, k + j)  # Cannot be A4[i, j]
+    assert A[n - i - 1, k - j - 1] == MatrixElement(A, n - i - 1, k - j - 1)  # Cannot be A1[n - i - 1, k - j - 1]
+
+
+def test_block_index_symbolic_nonzero():
+    # All invalid simplifications from test_block_index_symbolic() that become valid if all
+    # matrices have nonzero size and all indices are nonnegative
+    k, l, m, n = symbols('k l m n', integer=True, positive=True)
+    i, j = symbols('i j', integer=True, nonnegative=True)
+    A1 = MatrixSymbol('A1', n, k)
+    A2 = MatrixSymbol('A2', n, l)
+    A3 = MatrixSymbol('A3', m, k)
+    A4 = MatrixSymbol('A4', m, l)
+    A = BlockMatrix([[A1, A2], [A3, A4]])
+    assert A[0, 0] == A1[0, 0]
+    assert A[n + m - 1, 0] == A3[m - 1, 0]
+    assert A[0, k + l - 1] == A2[0, l - 1]
+    assert A[n + m - 1, k + l - 1] == A4[m - 1, l - 1]
+    assert A[i, j] == MatrixElement(A, i, j)
+    assert A[n + i, k + j] == A4[i, j]
+    assert A[n - i - 1, k - j - 1] == A1[n - i - 1, k - j - 1]
+    assert A[2 * n, 2 * k] == A4[n, k]
+
+
+def test_block_index_large():
+    n, m, k = symbols('n m k', integer=True, positive=True)
+    i = symbols('i', integer=True, nonnegative=True)
+    A1 = MatrixSymbol('A1', n, n)
+    A2 = MatrixSymbol('A2', n, m)
+    A3 = MatrixSymbol('A3', n, k)
+    A4 = MatrixSymbol('A4', m, n)
+    A5 = MatrixSymbol('A5', m, m)
+    A6 = MatrixSymbol('A6', m, k)
+    A7 = MatrixSymbol('A7', k, n)
+    A8 = MatrixSymbol('A8', k, m)
+    A9 = MatrixSymbol('A9', k, k)
+    A = BlockMatrix([[A1, A2, A3], [A4, A5, A6], [A7, A8, A9]])
+    assert A[n + i, n + i] == MatrixElement(A, n + i, n + i)
+
+
+@XFAIL
+def test_block_index_symbolic_fail():
+    # To make this work, symbolic matrix dimensions would need to be somehow assumed nonnegative
+    # even if the symbols aren't specified as such.  Then 2 * n < n would correctly evaluate to
+    # False in BlockMatrix._entry()
+    A1 = MatrixSymbol('A1', n, 1)
+    A2 = MatrixSymbol('A2', m, 1)
+    A = BlockMatrix([[A1], [A2]])
+    assert A[2 * n, 0] == A2[n, 0]
+
+
+def test_slicing():
+    A.as_explicit()[0, :]  # does not raise an error
+
+
+def test_errors():
+    raises(IndexError, lambda: Identity(2)[1, 2, 3, 4, 5])
+    raises(IndexError, lambda: Identity(2)[[1, 2, 3, 4, 5]])
+
+
+def test_matrix_expression_to_indices():
+    i, j = symbols("i, j")
+    i1, i2, i3 = symbols("i_1:4")
+
+    def replace_dummies(expr):
+        repl = {i: Symbol(i.name) for i in expr.atoms(Dummy)}
+        return expr.xreplace(repl)
+
+    expr = W*X*Z
+    assert replace_dummies(expr._entry(i, j)) == \
+        Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
+    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
+
+    expr = Z.T*X.T*W.T
+    assert replace_dummies(expr._entry(i, j)) == \
+        Sum(W[j, i2]*X[i2, i1]*Z[i1, i], (i1, 0, m-1), (i2, 0, l-1))
+    assert MatrixExpr.from_index_summation(expr._entry(i, j), i) == expr
+
+    expr = W*X*Z + W*Y*Z
+    assert replace_dummies(expr._entry(i, j)) == \
+        Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
+        Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
+    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
+
+    expr = 2*W*X*Z + 3*W*Y*Z
+    assert replace_dummies(expr._entry(i, j)) == \
+        2*Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
+        3*Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
+    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
+
+    expr = W*(X + Y)*Z
+    assert replace_dummies(expr._entry(i, j)) == \
+            Sum(W[i, i1]*(X[i1, i2] + Y[i1, i2])*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
+    assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
+
+    expr = A*B**2*A
+    #assert replace_dummies(expr._entry(i, j)) == \
+    #        Sum(A[i, i1]*B[i1, i2]*B[i2, i3]*A[i3, j], (i1, 0, 1), (i2, 0, 1), (i3, 0, 1))
+
+    # Check that different dummies are used in sub-multiplications:
+    expr = (X1*X2 + X2*X1)*X3
+    assert replace_dummies(expr._entry(i, j)) == \
+           Sum((Sum(X1[i, i2] * X2[i2, i1], (i2, 0, m - 1)) + Sum(X1[i3, i1] * X2[i, i3], (i3, 0, m - 1))) * X3[
+               i1, j], (i1, 0, m - 1))
+
+
+def test_matrix_expression_from_index_summation():
+    from sympy.abc import a,b,c,d
+    A = MatrixSymbol("A", k, k)
+    B = MatrixSymbol("B", k, k)
+    C = MatrixSymbol("C", k, k)
+    w1 = MatrixSymbol("w1", k, 1)
+
+    i0, i1, i2, i3, i4 = symbols("i0:5", cls=Dummy)
+
+    expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
+    assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
+    expr = Sum(W.T[b,a]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
+    assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
+    expr = Sum(A[b, a]*B[b, c]*C[c, d], (b, 0, k-1), (c, 0, k-1))
+    assert MatrixSymbol.from_index_summation(expr, a) == A.T*B*C
+    expr = Sum(A[b, a]*B[c, b]*C[c, d], (b, 0, k-1), (c, 0, k-1))
+    assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
+    expr = Sum(C[c, d]*A[b, a]*B[c, b], (b, 0, k-1), (c, 0, k-1))
+    assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
+    expr = Sum(A[a, b] + B[a, b], (a, 0, k-1), (b, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == OneMatrix(1, k)*A*OneMatrix(k, 1) + OneMatrix(1, k)*B*OneMatrix(k, 1)
+    expr = Sum(A[a, b]**2, (a, 0, k - 1), (b, 0, k - 1))
+    assert MatrixExpr.from_index_summation(expr, a) == Trace(A * A.T)
+    expr = Sum(A[a, b]**3, (a, 0, k - 1), (b, 0, k - 1))
+    assert MatrixExpr.from_index_summation(expr, a) == Trace(HadamardPower(A.T, 2) * A)
+    expr = Sum((A[a, b] + B[a, b])*C[b, c], (b, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == (A+B)*C
+    expr = Sum((A[a, b] + B[b, a])*C[b, c], (b, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == (A+B.T)*C
+    expr = Sum(A[a, b]*A[b, c]*A[c, d], (b, 0, k-1), (c, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == A**3
+    expr = Sum(A[a, b]*A[b, c]*B[c, d], (b, 0, k-1), (c, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == A**2*B
+
+    # Parse the trace of a matrix:
+
+    expr = Sum(A[a, a], (a, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, None) == trace(A)
+    expr = Sum(A[a, a]*B[b, c]*C[c, d], (a, 0, k-1), (c, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, b) == trace(A)*B*C
+
+    # Check wrong sum ranges (should raise an exception):
+
+    ## Case 1: 0 to m instead of 0 to m-1
+    expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m))
+    raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
+    ## Case 2: 1 to m-1 instead of 0 to m-1
+    expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 1, m-1))
+    raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
+
+    # Parse nested sums:
+    expr = Sum(A[a, b]*Sum(B[b, c]*C[c, d], (c, 0, k-1)), (b, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == A*B*C
+
+    # Test Kronecker delta:
+    expr = Sum(A[a, b]*KroneckerDelta(b, c)*B[c, d], (b, 0, k-1), (c, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, a) == A*B
+
+    expr = Sum(KroneckerDelta(i1, m)*KroneckerDelta(i2, n)*A[i, i1]*A[j, i2], (i1, 0, k-1), (i2, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, m) == ArrayTensorProduct(A.T, A)
+
+    # Test numbered indices:
+    expr = Sum(A[i1, i2]*w1[i2, 0], (i2, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*w1, i1, 0)
+
+    expr = Sum(A[i1, i2]*B[i2, 0], (i2, 0, k-1))
+    assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*B, i1, 0)

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_inverse.py

@@ -0,0 +1,62 @@
+from sympy.core import symbols, S
+from sympy.matrices.expressions import MatrixSymbol, Inverse, MatPow, ZeroMatrix, OneMatrix
+from sympy.matrices.common import NonInvertibleMatrixError, NonSquareMatrixError
+from sympy.matrices import eye, Identity
+from sympy.testing.pytest import raises
+from sympy.assumptions.ask import Q
+from sympy.assumptions.refine import refine
+
+n, m, l = symbols('n m l', integer=True)
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', m, l)
+C = MatrixSymbol('C', n, n)
+D = MatrixSymbol('D', n, n)
+E = MatrixSymbol('E', m, n)
+
+
+def test_inverse():
+    assert Inverse(C).args == (C, S.NegativeOne)
+    assert Inverse(C).shape == (n, n)
+    assert Inverse(A*E).shape == (n, n)
+    assert Inverse(E*A).shape == (m, m)
+    assert Inverse(C).inverse() == C
+    assert Inverse(Inverse(C)).doit() == C
+    assert isinstance(Inverse(Inverse(C)), Inverse)
+
+    assert Inverse(*Inverse(E*A).args) == Inverse(E*A)
+
+    assert C.inverse().inverse() == C
+
+    assert C.inverse()*C == Identity(C.rows)
+
+    assert Identity(n).inverse() == Identity(n)
+    assert (3*Identity(n)).inverse() == Identity(n)/3
+
+    # Simplifies Muls if possible (i.e. submatrices are square)
+    assert (C*D).inverse() == D.I*C.I
+    # But still works when not possible
+    assert isinstance((A*E).inverse(), Inverse)
+    assert Inverse(C*D).doit(inv_expand=False) == Inverse(C*D)
+
+    assert Inverse(eye(3)).doit() == eye(3)
+    assert Inverse(eye(3)).doit(deep=False) == eye(3)
+
+    assert OneMatrix(1, 1).I == Identity(1)
+    assert isinstance(OneMatrix(n, n).I, Inverse)
+
+def test_inverse_non_invertible():
+    raises(NonInvertibleMatrixError, lambda: ZeroMatrix(n, n).I)
+    raises(NonInvertibleMatrixError, lambda: OneMatrix(2, 2).I)
+
+def test_refine():
+    assert refine(C.I, Q.orthogonal(C)) == C.T
+
+
+def test_inverse_matpow_canonicalization():
+    A = MatrixSymbol('A', 3, 3)
+    assert Inverse(MatPow(A, 3)).doit() == MatPow(Inverse(A), 3).doit()
+
+
+def test_nonsquare_error():
+    A = MatrixSymbol('A', 3, 4)
+    raises(NonSquareMatrixError, lambda: Inverse(A))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_kronecker.py

@@ -0,0 +1,150 @@
+from sympy.core.mod import Mod
+from sympy.core.numbers import I
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.integers import floor
+from sympy.matrices.dense import (Matrix, eye)
+from sympy.matrices import MatrixSymbol, Identity
+from sympy.matrices.expressions import det, trace
+
+from sympy.matrices.expressions.kronecker import (KroneckerProduct,
+                                                  kronecker_product,
+                                                  combine_kronecker)
+
+
+mat1 = Matrix([[1, 2 * I], [1 + I, 3]])
+mat2 = Matrix([[2 * I, 3], [4 * I, 2]])
+
+i, j, k, n, m, o, p, x = symbols('i,j,k,n,m,o,p,x')
+Z = MatrixSymbol('Z', n, n)
+W = MatrixSymbol('W', m, m)
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', n, m)
+C = MatrixSymbol('C', m, k)
+
+
+def test_KroneckerProduct():
+    assert isinstance(KroneckerProduct(A, B), KroneckerProduct)
+    assert KroneckerProduct(A, B).subs(A, C) == KroneckerProduct(C, B)
+    assert KroneckerProduct(A, C).shape == (n*m, m*k)
+    assert (KroneckerProduct(A, C) + KroneckerProduct(-A, C)).is_ZeroMatrix
+    assert (KroneckerProduct(W, Z) * KroneckerProduct(W.I, Z.I)).is_Identity
+
+
+def test_KroneckerProduct_identity():
+    assert KroneckerProduct(Identity(m), Identity(n)) == Identity(m*n)
+    assert KroneckerProduct(eye(2), eye(3)) == eye(6)
+
+
+def test_KroneckerProduct_explicit():
+    X = MatrixSymbol('X', 2, 2)
+    Y = MatrixSymbol('Y', 2, 2)
+    kp = KroneckerProduct(X, Y)
+    assert kp.shape == (4, 4)
+    assert kp.as_explicit() == Matrix(
+        [
+            [X[0, 0]*Y[0, 0], X[0, 0]*Y[0, 1], X[0, 1]*Y[0, 0], X[0, 1]*Y[0, 1]],
+            [X[0, 0]*Y[1, 0], X[0, 0]*Y[1, 1], X[0, 1]*Y[1, 0], X[0, 1]*Y[1, 1]],
+            [X[1, 0]*Y[0, 0], X[1, 0]*Y[0, 1], X[1, 1]*Y[0, 0], X[1, 1]*Y[0, 1]],
+            [X[1, 0]*Y[1, 0], X[1, 0]*Y[1, 1], X[1, 1]*Y[1, 0], X[1, 1]*Y[1, 1]]
+        ]
+    )
+
+
+def test_tensor_product_adjoint():
+    assert KroneckerProduct(I*A, B).adjoint() == \
+        -I*KroneckerProduct(A.adjoint(), B.adjoint())
+    assert KroneckerProduct(mat1, mat2).adjoint() == \
+        kronecker_product(mat1.adjoint(), mat2.adjoint())
+
+
+def test_tensor_product_conjugate():
+    assert KroneckerProduct(I*A, B).conjugate() == \
+        -I*KroneckerProduct(A.conjugate(), B.conjugate())
+    assert KroneckerProduct(mat1, mat2).conjugate() == \
+        kronecker_product(mat1.conjugate(), mat2.conjugate())
+
+
+def test_tensor_product_transpose():
+    assert KroneckerProduct(I*A, B).transpose() == \
+        I*KroneckerProduct(A.transpose(), B.transpose())
+    assert KroneckerProduct(mat1, mat2).transpose() == \
+        kronecker_product(mat1.transpose(), mat2.transpose())
+
+
+def test_KroneckerProduct_is_associative():
+    assert kronecker_product(A, kronecker_product(
+        B, C)) == kronecker_product(kronecker_product(A, B), C)
+    assert kronecker_product(A, kronecker_product(
+        B, C)) == KroneckerProduct(A, B, C)
+
+
+def test_KroneckerProduct_is_bilinear():
+    assert kronecker_product(x*A, B) == x*kronecker_product(A, B)
+    assert kronecker_product(A, x*B) == x*kronecker_product(A, B)
+
+
+def test_KroneckerProduct_determinant():
+    kp = kronecker_product(W, Z)
+    assert det(kp) == det(W)**n * det(Z)**m
+
+
+def test_KroneckerProduct_trace():
+    kp = kronecker_product(W, Z)
+    assert trace(kp) == trace(W)*trace(Z)
+
+
+def test_KroneckerProduct_isnt_commutative():
+    assert KroneckerProduct(A, B) != KroneckerProduct(B, A)
+    assert KroneckerProduct(A, B).is_commutative is False
+
+
+def test_KroneckerProduct_extracts_commutative_part():
+    assert kronecker_product(x * A, 2 * B) == x * \
+        2 * KroneckerProduct(A, B)
+
+
+def test_KroneckerProduct_inverse():
+    kp = kronecker_product(W, Z)
+    assert kp.inverse() == kronecker_product(W.inverse(), Z.inverse())
+
+
+def test_KroneckerProduct_combine_add():
+    kp1 = kronecker_product(A, B)
+    kp2 = kronecker_product(C, W)
+    assert combine_kronecker(kp1*kp2) == kronecker_product(A*C, B*W)
+
+
+def test_KroneckerProduct_combine_mul():
+    X = MatrixSymbol('X', m, n)
+    Y = MatrixSymbol('Y', m, n)
+    kp1 = kronecker_product(A, X)
+    kp2 = kronecker_product(B, Y)
+    assert combine_kronecker(kp1+kp2) == kronecker_product(A+B, X+Y)
+
+
+def test_KroneckerProduct_combine_pow():
+    X = MatrixSymbol('X', n, n)
+    Y = MatrixSymbol('Y', n, n)
+    assert combine_kronecker(KroneckerProduct(
+        X, Y)**x) == KroneckerProduct(X**x, Y**x)
+    assert combine_kronecker(x * KroneckerProduct(X, Y)
+                             ** 2) == x * KroneckerProduct(X**2, Y**2)
+    assert combine_kronecker(
+        x * (KroneckerProduct(X, Y)**2) * KroneckerProduct(A, B)) == x * KroneckerProduct(X**2 * A, Y**2 * B)
+    # cannot simplify because of non-square arguments to kronecker product:
+    assert combine_kronecker(KroneckerProduct(A, B.T) ** m) == KroneckerProduct(A, B.T) ** m
+
+
+def test_KroneckerProduct_expand():
+    X = MatrixSymbol('X', n, n)
+    Y = MatrixSymbol('Y', n, n)
+
+    assert KroneckerProduct(X + Y, Y + Z).expand(kroneckerproduct=True) == \
+        KroneckerProduct(X, Y) + KroneckerProduct(X, Z) + \
+        KroneckerProduct(Y, Y) + KroneckerProduct(Y, Z)
+
+def test_KroneckerProduct_entry():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', o, p)
+
+    assert KroneckerProduct(A, B)._entry(i, j) == A[Mod(floor(i/o), n), Mod(floor(j/p), m)]*B[Mod(i, o), Mod(j, p)]

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_matadd.py

@@ -0,0 +1,58 @@
+from sympy.matrices.expressions import MatrixSymbol, MatAdd, MatPow, MatMul
+from sympy.matrices.expressions.special import GenericZeroMatrix, ZeroMatrix
+from sympy.matrices.common import ShapeError
+from sympy.matrices import eye, ImmutableMatrix
+from sympy.core import Add, Basic, S
+from sympy.core.add import add
+from sympy.testing.pytest import XFAIL, raises
+
+X = MatrixSymbol('X', 2, 2)
+Y = MatrixSymbol('Y', 2, 2)
+
+def test_evaluate():
+    assert MatAdd(X, X, evaluate=True) == add(X, X, evaluate=True) == MatAdd(X, X).doit()
+
+def test_sort_key():
+    assert MatAdd(Y, X).doit().args == add(Y, X).doit().args == (X, Y)
+
+
+def test_matadd_sympify():
+    assert isinstance(MatAdd(eye(1), eye(1)).args[0], Basic)
+    assert isinstance(add(eye(1), eye(1)).args[0], Basic)
+
+
+def test_matadd_of_matrices():
+    assert MatAdd(eye(2), 4*eye(2), eye(2)).doit() == ImmutableMatrix(6*eye(2))
+    assert add(eye(2), 4*eye(2), eye(2)).doit() == ImmutableMatrix(6*eye(2))
+
+
+def test_doit_args():
+    A = ImmutableMatrix([[1, 2], [3, 4]])
+    B = ImmutableMatrix([[2, 3], [4, 5]])
+    assert MatAdd(A, MatPow(B, 2)).doit() == A + B**2
+    assert MatAdd(A, MatMul(A, B)).doit() == A + A*B
+    assert (MatAdd(A, X, MatMul(A, B), Y, MatAdd(2*A, B)).doit() ==
+    add(A, X, MatMul(A, B), Y, add(2*A, B)).doit() ==
+    MatAdd(3*A + A*B + B, X, Y))
+
+
+def test_generic_identity():
+    assert MatAdd.identity == GenericZeroMatrix()
+    assert MatAdd.identity != S.Zero
+
+
+def test_zero_matrix_add():
+    assert Add(ZeroMatrix(2, 2), ZeroMatrix(2, 2)) == ZeroMatrix(2, 2)
+
+@XFAIL
+def test_matrix_Add_with_scalar():
+    raises(TypeError, lambda: Add(0, ZeroMatrix(2, 2)))
+
+
+def test_shape_error():
+    A = MatrixSymbol('A', 2, 3)
+    B = MatrixSymbol('B', 3, 3)
+    raises(ShapeError, lambda: MatAdd(A, B))
+
+    A = MatrixSymbol('A', 3, 2)
+    raises(ShapeError, lambda: MatAdd(A, B))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_matexpr.py

@@ -0,0 +1,567 @@
+from sympy.concrete.summations import Sum
+from sympy.core.exprtools import gcd_terms
+from sympy.core.function import (diff, expand)
+from sympy.core.relational import Eq
+from sympy.core.symbol import (Dummy, Symbol, Str)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.matrices.dense import zeros
+from sympy.polys.polytools import factor
+
+from sympy.core import (S, symbols, Add, Mul, SympifyError, Rational,
+                    Function)
+from sympy.functions import sin, cos, tan, sqrt, cbrt, exp
+from sympy.simplify import simplify
+from sympy.matrices import (ImmutableMatrix, Inverse, MatAdd, MatMul,
+        MatPow, Matrix, MatrixExpr, MatrixSymbol,
+        SparseMatrix, Transpose, Adjoint, MatrixSet)
+from sympy.matrices.common import NonSquareMatrixError
+from sympy.matrices.expressions.determinant import Determinant, det
+from sympy.matrices.expressions.matexpr import MatrixElement
+from sympy.matrices.expressions.special import ZeroMatrix, Identity
+from sympy.testing.pytest import raises, XFAIL
+
+
+n, m, l, k, p = symbols('n m l k p', integer=True)
+x = symbols('x')
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', m, l)
+C = MatrixSymbol('C', n, n)
+D = MatrixSymbol('D', n, n)
+E = MatrixSymbol('E', m, n)
+w = MatrixSymbol('w', n, 1)
+
+
+def test_matrix_symbol_creation():
+    assert MatrixSymbol('A', 2, 2)
+    assert MatrixSymbol('A', 0, 0)
+    raises(ValueError, lambda: MatrixSymbol('A', -1, 2))
+    raises(ValueError, lambda: MatrixSymbol('A', 2.0, 2))
+    raises(ValueError, lambda: MatrixSymbol('A', 2j, 2))
+    raises(ValueError, lambda: MatrixSymbol('A', 2, -1))
+    raises(ValueError, lambda: MatrixSymbol('A', 2, 2.0))
+    raises(ValueError, lambda: MatrixSymbol('A', 2, 2j))
+
+    n = symbols('n')
+    assert MatrixSymbol('A', n, n)
+    n = symbols('n', integer=False)
+    raises(ValueError, lambda: MatrixSymbol('A', n, n))
+    n = symbols('n', negative=True)
+    raises(ValueError, lambda: MatrixSymbol('A', n, n))
+
+
+def test_matexpr_properties():
+    assert A.shape == (n, m)
+    assert (A * B).shape == (n, l)
+    assert A[0, 1].indices == (0, 1)
+    assert A[0, 0].symbol == A
+    assert A[0, 0].symbol.name == 'A'
+
+
+def test_matexpr():
+    assert (x*A).shape == A.shape
+    assert (x*A).__class__ == MatMul
+    assert 2*A - A - A == ZeroMatrix(*A.shape)
+    assert (A*B).shape == (n, l)
+
+
+def test_matexpr_subs():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', m, l)
+    C = MatrixSymbol('C', m, l)
+
+    assert A.subs(n, m).shape == (m, m)
+    assert (A*B).subs(B, C) == A*C
+    assert (A*B).subs(l, n).is_square
+
+    W = MatrixSymbol("W", 3, 3)
+    X = MatrixSymbol("X", 2, 2)
+    Y = MatrixSymbol("Y", 1, 2)
+    Z = MatrixSymbol("Z", n, 2)
+    # no restrictions on Symbol replacement
+    assert X.subs(X, Y) == Y
+    # it might be better to just change the name
+    y = Str('y')
+    assert X.subs(Str("X"), y).args == (y, 2, 2)
+    # it's ok to introduce a wider matrix
+    assert X[1, 1].subs(X, W) == W[1, 1]
+    # but for a given MatrixExpression, only change
+    # name if indexing on the new shape is valid.
+    # Here, X is 2,2; Y is 1,2 and Y[1, 1] is out
+    # of range so an error is raised
+    raises(IndexError, lambda: X[1, 1].subs(X, Y))
+    # here, [0, 1] is in range so the subs succeeds
+    assert X[0, 1].subs(X, Y) == Y[0, 1]
+    # and here the size of n will accept any index
+    # in the first position
+    assert W[2, 1].subs(W, Z) == Z[2, 1]
+    # but not in the second position
+    raises(IndexError, lambda: W[2, 2].subs(W, Z))
+    # any matrix should raise if invalid
+    raises(IndexError, lambda: W[2, 2].subs(W, zeros(2)))
+
+    A = SparseMatrix([[1, 2], [3, 4]])
+    B = Matrix([[1, 2], [3, 4]])
+    C, D = MatrixSymbol('C', 2, 2), MatrixSymbol('D', 2, 2)
+
+    assert (C*D).subs({C: A, D: B}) == MatMul(A, B)
+
+
+def test_addition():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', n, m)
+
+    assert isinstance(A + B, MatAdd)
+    assert (A + B).shape == A.shape
+    assert isinstance(A - A + 2*B, MatMul)
+
+    raises(TypeError, lambda: A + 1)
+    raises(TypeError, lambda: 5 + A)
+    raises(TypeError, lambda: 5 - A)
+
+    assert A + ZeroMatrix(n, m) - A == ZeroMatrix(n, m)
+    raises(TypeError, lambda: ZeroMatrix(n, m) + S.Zero)
+
+
+def test_multiplication():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', m, l)
+    C = MatrixSymbol('C', n, n)
+
+    assert (2*A*B).shape == (n, l)
+    assert (A*0*B) == ZeroMatrix(n, l)
+    assert (2*A).shape == A.shape
+
+    assert A * ZeroMatrix(m, m) * B == ZeroMatrix(n, l)
+
+    assert C * Identity(n) * C.I == Identity(n)
+
+    assert B/2 == S.Half*B
+    raises(NotImplementedError, lambda: 2/B)
+
+    A = MatrixSymbol('A', n, n)
+    B = MatrixSymbol('B', n, n)
+    assert Identity(n) * (A + B) == A + B
+
+    assert A**2*A == A**3
+    assert A**2*(A.I)**3 == A.I
+    assert A**3*(A.I)**2 == A
+
+
+def test_MatPow():
+    A = MatrixSymbol('A', n, n)
+
+    AA = MatPow(A, 2)
+    assert AA.exp == 2
+    assert AA.base == A
+    assert (A**n).exp == n
+
+    assert A**0 == Identity(n)
+    assert A**1 == A
+    assert A**2 == AA
+    assert A**-1 == Inverse(A)
+    assert (A**-1)**-1 == A
+    assert (A**2)**3 == A**6
+    assert A**S.Half == sqrt(A)
+    assert A**Rational(1, 3) == cbrt(A)
+    raises(NonSquareMatrixError, lambda: MatrixSymbol('B', 3, 2)**2)
+
+
+def test_MatrixSymbol():
+    n, m, t = symbols('n,m,t')
+    X = MatrixSymbol('X', n, m)
+    assert X.shape == (n, m)
+    raises(TypeError, lambda: MatrixSymbol('X', n, m)(t))  # issue 5855
+    assert X.doit() == X
+
+
+def test_dense_conversion():
+    X = MatrixSymbol('X', 2, 2)
+    assert ImmutableMatrix(X) == ImmutableMatrix(2, 2, lambda i, j: X[i, j])
+    assert Matrix(X) == Matrix(2, 2, lambda i, j: X[i, j])
+
+
+def test_free_symbols():
+    assert (C*D).free_symbols == {C, D}
+
+
+def test_zero_matmul():
+    assert isinstance(S.Zero * MatrixSymbol('X', 2, 2), MatrixExpr)
+
+
+def test_matadd_simplify():
+    A = MatrixSymbol('A', 1, 1)
+    assert simplify(MatAdd(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
+        MatAdd(A, Matrix([[1]]))
+
+
+def test_matmul_simplify():
+    A = MatrixSymbol('A', 1, 1)
+    assert simplify(MatMul(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
+        MatMul(A, Matrix([[1]]))
+
+
+def test_invariants():
+    A = MatrixSymbol('A', n, m)
+    B = MatrixSymbol('B', m, l)
+    X = MatrixSymbol('X', n, n)
+    objs = [Identity(n), ZeroMatrix(m, n), A, MatMul(A, B), MatAdd(A, A),
+            Transpose(A), Adjoint(A), Inverse(X), MatPow(X, 2), MatPow(X, -1),
+            MatPow(X, 0)]
+    for obj in objs:
+        assert obj == obj.__class__(*obj.args)
+
+
+def test_matexpr_indexing():
+    A = MatrixSymbol('A', n, m)
+    A[1, 2]
+    A[l, k]
+    A[l + 1, k + 1]
+    A = MatrixSymbol('A', 2, 1)
+    for i in range(-2, 2):
+        for j in range(-1, 1):
+            A[i, j]
+
+
+def test_single_indexing():
+    A = MatrixSymbol('A', 2, 3)
+    assert A[1] == A[0, 1]
+    assert A[int(1)] == A[0, 1]
+    assert A[3] == A[1, 0]
+    assert list(A[:2, :2]) == [A[0, 0], A[0, 1], A[1, 0], A[1, 1]]
+    raises(IndexError, lambda: A[6])
+    raises(IndexError, lambda: A[n])
+    B = MatrixSymbol('B', n, m)
+    raises(IndexError, lambda: B[1])
+    B = MatrixSymbol('B', n, 3)
+    assert B[3] == B[1, 0]
+
+
+def test_MatrixElement_commutative():
+    assert A[0, 1]*A[1, 0] == A[1, 0]*A[0, 1]
+
+
+def test_MatrixSymbol_determinant():
+    A = MatrixSymbol('A', 4, 4)
+    assert A.as_explicit().det() == A[0, 0]*A[1, 1]*A[2, 2]*A[3, 3] - \
+        A[0, 0]*A[1, 1]*A[2, 3]*A[3, 2] - A[0, 0]*A[1, 2]*A[2, 1]*A[3, 3] + \
+        A[0, 0]*A[1, 2]*A[2, 3]*A[3, 1] + A[0, 0]*A[1, 3]*A[2, 1]*A[3, 2] - \
+        A[0, 0]*A[1, 3]*A[2, 2]*A[3, 1] - A[0, 1]*A[1, 0]*A[2, 2]*A[3, 3] + \
+        A[0, 1]*A[1, 0]*A[2, 3]*A[3, 2] + A[0, 1]*A[1, 2]*A[2, 0]*A[3, 3] - \
+        A[0, 1]*A[1, 2]*A[2, 3]*A[3, 0] - A[0, 1]*A[1, 3]*A[2, 0]*A[3, 2] + \
+        A[0, 1]*A[1, 3]*A[2, 2]*A[3, 0] + A[0, 2]*A[1, 0]*A[2, 1]*A[3, 3] - \
+        A[0, 2]*A[1, 0]*A[2, 3]*A[3, 1] - A[0, 2]*A[1, 1]*A[2, 0]*A[3, 3] + \
+        A[0, 2]*A[1, 1]*A[2, 3]*A[3, 0] + A[0, 2]*A[1, 3]*A[2, 0]*A[3, 1] - \
+        A[0, 2]*A[1, 3]*A[2, 1]*A[3, 0] - A[0, 3]*A[1, 0]*A[2, 1]*A[3, 2] + \
+        A[0, 3]*A[1, 0]*A[2, 2]*A[3, 1] + A[0, 3]*A[1, 1]*A[2, 0]*A[3, 2] - \
+        A[0, 3]*A[1, 1]*A[2, 2]*A[3, 0] - A[0, 3]*A[1, 2]*A[2, 0]*A[3, 1] + \
+        A[0, 3]*A[1, 2]*A[2, 1]*A[3, 0]
+
+    B = MatrixSymbol('B', 4, 4)
+    assert Determinant(A + B).doit() == det(A + B) == (A + B).det()
+
+
+def test_MatrixElement_diff():
+    assert (A[3, 0]*A[0, 0]).diff(A[0, 0]) == A[3, 0]
+
+
+def test_MatrixElement_doit():
+    u = MatrixSymbol('u', 2, 1)
+    v = ImmutableMatrix([3, 5])
+    assert u[0, 0].subs(u, v).doit() == v[0, 0]
+
+
+def test_identity_powers():
+    M = Identity(n)
+    assert MatPow(M, 3).doit() == M**3
+    assert M**n == M
+    assert MatPow(M, 0).doit() == M**2
+    assert M**-2 == M
+    assert MatPow(M, -2).doit() == M**0
+    N = Identity(3)
+    assert MatPow(N, 2).doit() == N**n
+    assert MatPow(N, 3).doit() == N
+    assert MatPow(N, -2).doit() == N**4
+    assert MatPow(N, 2).doit() == N**0
+
+
+def test_Zero_power():
+    z1 = ZeroMatrix(n, n)
+    assert z1**4 == z1
+    raises(ValueError, lambda:z1**-2)
+    assert z1**0 == Identity(n)
+    assert MatPow(z1, 2).doit() == z1**2
+    raises(ValueError, lambda:MatPow(z1, -2).doit())
+    z2 = ZeroMatrix(3, 3)
+    assert MatPow(z2, 4).doit() == z2**4
+    raises(ValueError, lambda:z2**-3)
+    assert z2**3 == MatPow(z2, 3).doit()
+    assert z2**0 == Identity(3)
+    raises(ValueError, lambda:MatPow(z2, -1).doit())
+
+
+def test_matrixelement_diff():
+    dexpr = diff((D*w)[k,0], w[p,0])
+
+    assert w[k, p].diff(w[k, p]) == 1
+    assert w[k, p].diff(w[0, 0]) == KroneckerDelta(0, k, (0, n-1))*KroneckerDelta(0, p, (0, 0))
+    _i_1 = Dummy("_i_1")
+    assert dexpr.dummy_eq(Sum(KroneckerDelta(_i_1, p, (0, n-1))*D[k, _i_1], (_i_1, 0, n - 1)))
+    assert dexpr.doit() == D[k, p]
+
+
+def test_MatrixElement_with_values():
+    x, y, z, w = symbols("x y z w")
+    M = Matrix([[x, y], [z, w]])
+    i, j = symbols("i, j")
+    Mij = M[i, j]
+    assert isinstance(Mij, MatrixElement)
+    Ms = SparseMatrix([[2, 3], [4, 5]])
+    msij = Ms[i, j]
+    assert isinstance(msij, MatrixElement)
+    for oi, oj in [(0, 0), (0, 1), (1, 0), (1, 1)]:
+        assert Mij.subs({i: oi, j: oj}) == M[oi, oj]
+        assert msij.subs({i: oi, j: oj}) == Ms[oi, oj]
+    A = MatrixSymbol("A", 2, 2)
+    assert A[0, 0].subs(A, M) == x
+    assert A[i, j].subs(A, M) == M[i, j]
+    assert M[i, j].subs(M, A) == A[i, j]
+
+    assert isinstance(M[3*i - 2, j], MatrixElement)
+    assert M[3*i - 2, j].subs({i: 1, j: 0}) == M[1, 0]
+    assert isinstance(M[i, 0], MatrixElement)
+    assert M[i, 0].subs(i, 0) == M[0, 0]
+    assert M[0, i].subs(i, 1) == M[0, 1]
+
+    assert M[i, j].diff(x) == Matrix([[1, 0], [0, 0]])[i, j]
+
+    raises(ValueError, lambda: M[i, 2])
+    raises(ValueError, lambda: M[i, -1])
+    raises(ValueError, lambda: M[2, i])
+    raises(ValueError, lambda: M[-1, i])
+
+
+def test_inv():
+    B = MatrixSymbol('B', 3, 3)
+    assert B.inv() == B**-1
+
+    # https://github.com/sympy/sympy/issues/19162
+    X = MatrixSymbol('X', 1, 1).as_explicit()
+    assert X.inv() == Matrix([[1/X[0, 0]]])
+
+    X = MatrixSymbol('X', 2, 2).as_explicit()
+    detX = X[0, 0]*X[1, 1] - X[0, 1]*X[1, 0]
+    invX = Matrix([[ X[1, 1], -X[0, 1]],
+                   [-X[1, 0],  X[0, 0]]]) / detX
+    assert X.inv() == invX
+
+
+@XFAIL
+def test_factor_expand():
+    A = MatrixSymbol("A", n, n)
+    B = MatrixSymbol("B", n, n)
+    expr1 = (A + B)*(C + D)
+    expr2 = A*C + B*C + A*D + B*D
+    assert expr1 != expr2
+    assert expand(expr1) == expr2
+    assert factor(expr2) == expr1
+
+    expr = B**(-1)*(A**(-1)*B**(-1) - A**(-1)*C*B**(-1))**(-1)*A**(-1)
+    I = Identity(n)
+    # Ideally we get the first, but we at least don't want a wrong answer
+    assert factor(expr) in [I - C, B**-1*(A**-1*(I - C)*B**-1)**-1*A**-1]
+
+
+def test_issue_2749():
+    A = MatrixSymbol("A", 5, 2)
+    assert (A.T * A).I.as_explicit() == Matrix([[(A.T * A).I[0, 0], (A.T * A).I[0, 1]], \
+    [(A.T * A).I[1, 0], (A.T * A).I[1, 1]]])
+
+
+def test_issue_2750():
+    x = MatrixSymbol('x', 1, 1)
+    assert (x.T*x).as_explicit()**-1 == Matrix([[x[0, 0]**(-2)]])
+
+
+def test_issue_7842():
+    A = MatrixSymbol('A', 3, 1)
+    B = MatrixSymbol('B', 2, 1)
+    assert Eq(A, B) == False
+    assert Eq(A[1,0], B[1, 0]).func is Eq
+    A = ZeroMatrix(2, 3)
+    B = ZeroMatrix(2, 3)
+    assert Eq(A, B) == True
+
+
+def test_issue_21195():
+    t = symbols('t')
+    x = Function('x')(t)
+    dx = x.diff(t)
+    exp1 = cos(x) + cos(x)*dx
+    exp2 = sin(x) + tan(x)*(dx.diff(t))
+    exp3 = sin(x)*sin(t)*(dx.diff(t)).diff(t)
+    A = Matrix([[exp1], [exp2], [exp3]])
+    B = Matrix([[exp1.diff(x)], [exp2.diff(x)], [exp3.diff(x)]])
+    assert A.diff(x) == B
+
+
+def test_MatMul_postprocessor():
+    z = zeros(2)
+    z1 = ZeroMatrix(2, 2)
+    assert Mul(0, z) == Mul(z, 0) in [z, z1]
+
+    M = Matrix([[1, 2], [3, 4]])
+    Mx = Matrix([[x, 2*x], [3*x, 4*x]])
+    assert Mul(x, M) == Mul(M, x) == Mx
+
+    A = MatrixSymbol("A", 2, 2)
+    assert Mul(A, M) == MatMul(A, M)
+    assert Mul(M, A) == MatMul(M, A)
+    # Scalars should be absorbed into constant matrices
+    a = Mul(x, M, A)
+    b = Mul(M, x, A)
+    c = Mul(M, A, x)
+    assert a == b == c == MatMul(Mx, A)
+    a = Mul(x, A, M)
+    b = Mul(A, x, M)
+    c = Mul(A, M, x)
+    assert a == b == c == MatMul(A, Mx)
+    assert Mul(M, M) == M**2
+    assert Mul(A, M, M) == MatMul(A, M**2)
+    assert Mul(M, M, A) == MatMul(M**2, A)
+    assert Mul(M, A, M) == MatMul(M, A, M)
+
+    assert Mul(A, x, M, M, x) == MatMul(A, Mx**2)
+
+
+@XFAIL
+def test_MatAdd_postprocessor_xfail():
+    # This is difficult to get working because of the way that Add processes
+    # its args.
+    z = zeros(2)
+    assert Add(z, S.NaN) == Add(S.NaN, z)
+
+
+def test_MatAdd_postprocessor():
+    # Some of these are nonsensical, but we do not raise errors for Add
+    # because that breaks algorithms that want to replace matrices with dummy
+    # symbols.
+
+    z = zeros(2)
+
+    assert Add(0, z) == Add(z, 0) == z
+
+    a = Add(S.Infinity, z)
+    assert a == Add(z, S.Infinity)
+    assert isinstance(a, Add)
+    assert a.args == (S.Infinity, z)
+
+    a = Add(S.ComplexInfinity, z)
+    assert a == Add(z, S.ComplexInfinity)
+    assert isinstance(a, Add)
+    assert a.args == (S.ComplexInfinity, z)
+
+    a = Add(z, S.NaN)
+    # assert a == Add(S.NaN, z) # See the XFAIL above
+    assert isinstance(a, Add)
+    assert a.args == (S.NaN, z)
+
+    M = Matrix([[1, 2], [3, 4]])
+    a = Add(x, M)
+    assert a == Add(M, x)
+    assert isinstance(a, Add)
+    assert a.args == (x, M)
+
+    A = MatrixSymbol("A", 2, 2)
+    assert Add(A, M) == Add(M, A) == A + M
+
+    # Scalars should be absorbed into constant matrices (producing an error)
+    a = Add(x, M, A)
+    assert a == Add(M, x, A) == Add(M, A, x) == Add(x, A, M) == Add(A, x, M) == Add(A, M, x)
+    assert isinstance(a, Add)
+    assert a.args == (x, A + M)
+
+    assert Add(M, M) == 2*M
+    assert Add(M, A, M) == Add(M, M, A) == Add(A, M, M) == A + 2*M
+
+    a = Add(A, x, M, M, x)
+    assert isinstance(a, Add)
+    assert a.args == (2*x, A + 2*M)
+
+
+def test_simplify_matrix_expressions():
+    # Various simplification functions
+    assert type(gcd_terms(C*D + D*C)) == MatAdd
+    a = gcd_terms(2*C*D + 4*D*C)
+    assert type(a) == MatAdd
+    assert a.args == (2*C*D, 4*D*C)
+
+
+def test_exp():
+    A = MatrixSymbol('A', 2, 2)
+    B = MatrixSymbol('B', 2, 2)
+    expr1 = exp(A)*exp(B)
+    expr2 = exp(B)*exp(A)
+    assert expr1 != expr2
+    assert expr1 - expr2 != 0
+    assert not isinstance(expr1, exp)
+    assert not isinstance(expr2, exp)
+
+
+def test_invalid_args():
+    raises(SympifyError, lambda: MatrixSymbol(1, 2, 'A'))
+
+
+def test_matrixsymbol_from_symbol():
+    # The label should be preserved during doit and subs
+    A_label = Symbol('A', complex=True)
+    A = MatrixSymbol(A_label, 2, 2)
+
+    A_1 = A.doit()
+    A_2 = A.subs(2, 3)
+    assert A_1.args == A.args
+    assert A_2.args[0] == A.args[0]
+
+
+def test_as_explicit():
+    Z = MatrixSymbol('Z', 2, 3)
+    assert Z.as_explicit() == ImmutableMatrix([
+        [Z[0, 0], Z[0, 1], Z[0, 2]],
+        [Z[1, 0], Z[1, 1], Z[1, 2]],
+    ])
+    raises(ValueError, lambda: A.as_explicit())
+
+
+def test_MatrixSet():
+    M = MatrixSet(2, 2, set=S.Reals)
+    assert M.shape == (2, 2)
+    assert M.set == S.Reals
+    X = Matrix([[1, 2], [3, 4]])
+    assert X in M
+    X = ZeroMatrix(2, 2)
+    assert X in M
+    raises(TypeError, lambda: A in M)
+    raises(TypeError, lambda: 1 in M)
+    M = MatrixSet(n, m, set=S.Reals)
+    assert A in M
+    raises(TypeError, lambda: C in M)
+    raises(TypeError, lambda: X in M)
+    M = MatrixSet(2, 2, set={1, 2, 3})
+    X = Matrix([[1, 2], [3, 4]])
+    Y = Matrix([[1, 2]])
+    assert (X in M) == S.false
+    assert (Y in M) == S.false
+    raises(ValueError, lambda: MatrixSet(2, -2, S.Reals))
+    raises(ValueError, lambda: MatrixSet(2.4, -1, S.Reals))
+    raises(TypeError, lambda: MatrixSet(2, 2, (1, 2, 3)))
+
+
+def test_matrixsymbol_solving():
+    A = MatrixSymbol('A', 2, 2)
+    B = MatrixSymbol('B', 2, 2)
+    Z = ZeroMatrix(2, 2)
+    assert -(-A + B) - A + B == Z
+    assert (-(-A + B) - A + B).simplify() == Z
+    assert (-(-A + B) - A + B).expand() == Z
+    assert (-(-A + B) - A + B - Z).simplify() == Z
+    assert (-(-A + B) - A + B - Z).expand() == Z
+    assert (A*(A + B) + B*(A.T + B.T)).expand() == A**2 + A*B + B*A.T + B*B.T

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_matmul.py

@@ -0,0 +1,186 @@
+from sympy.core import I, symbols, Basic, Mul, S
+from sympy.core.mul import mul
+from sympy.functions import adjoint, transpose
+from sympy.matrices.common import ShapeError
+from sympy.matrices import (Identity, Inverse, Matrix, MatrixSymbol, ZeroMatrix,
+        eye, ImmutableMatrix)
+from sympy.matrices.expressions import Adjoint, Transpose, det, MatPow
+from sympy.matrices.expressions.special import GenericIdentity
+from sympy.matrices.expressions.matmul import (factor_in_front, remove_ids,
+        MatMul, combine_powers, any_zeros, unpack, only_squares)
+from sympy.strategies import null_safe
+from sympy.assumptions.ask import Q
+from sympy.assumptions.refine import refine
+from sympy.core.symbol import Symbol
+
+from sympy.testing.pytest import XFAIL, raises
+
+n, m, l, k = symbols('n m l k', integer=True)
+x = symbols('x')
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', m, l)
+C = MatrixSymbol('C', n, n)
+D = MatrixSymbol('D', n, n)
+E = MatrixSymbol('E', m, n)
+
+def test_evaluate():
+    assert MatMul(C, C, evaluate=True) == MatMul(C, C).doit()
+
+def test_adjoint():
+    assert adjoint(A*B) == Adjoint(B)*Adjoint(A)
+    assert adjoint(2*A*B) == 2*Adjoint(B)*Adjoint(A)
+    assert adjoint(2*I*C) == -2*I*Adjoint(C)
+
+    M = Matrix(2, 2, [1, 2 + I, 3, 4])
+    MA = Matrix(2, 2, [1, 3, 2 - I, 4])
+    assert adjoint(M) == MA
+    assert adjoint(2*M) == 2*MA
+    assert adjoint(MatMul(2, M)) == MatMul(2, MA).doit()
+
+
+def test_transpose():
+    assert transpose(A*B) == Transpose(B)*Transpose(A)
+    assert transpose(2*A*B) == 2*Transpose(B)*Transpose(A)
+    assert transpose(2*I*C) == 2*I*Transpose(C)
+
+    M = Matrix(2, 2, [1, 2 + I, 3, 4])
+    MT = Matrix(2, 2, [1, 3, 2 + I, 4])
+    assert transpose(M) == MT
+    assert transpose(2*M) == 2*MT
+    assert transpose(x*M) == x*MT
+    assert transpose(MatMul(2, M)) == MatMul(2, MT).doit()
+
+
+def test_factor_in_front():
+    assert factor_in_front(MatMul(A, 2, B, evaluate=False)) ==\
+                           MatMul(2, A, B, evaluate=False)
+
+
+def test_remove_ids():
+    assert remove_ids(MatMul(A, Identity(m), B, evaluate=False)) == \
+                      MatMul(A, B, evaluate=False)
+    assert null_safe(remove_ids)(MatMul(Identity(n), evaluate=False)) == \
+                                 MatMul(Identity(n), evaluate=False)
+
+
+def test_combine_powers():
+    assert combine_powers(MatMul(D, Inverse(D), D, evaluate=False)) == \
+                 MatMul(Identity(n), D, evaluate=False)
+    assert combine_powers(MatMul(B.T, Inverse(E*A), E, A, B, evaluate=False)) == \
+        MatMul(B.T, Identity(m), B, evaluate=False)
+    assert combine_powers(MatMul(A, E, Inverse(A*E), D, evaluate=False)) == \
+        MatMul(Identity(n), D, evaluate=False)
+
+
+def test_any_zeros():
+    assert any_zeros(MatMul(A, ZeroMatrix(m, k), evaluate=False)) == \
+                     ZeroMatrix(n, k)
+
+
+def test_unpack():
+    assert unpack(MatMul(A, evaluate=False)) == A
+    x = MatMul(A, B)
+    assert unpack(x) == x
+
+
+def test_only_squares():
+    assert only_squares(C) == [C]
+    assert only_squares(C, D) == [C, D]
+    assert only_squares(C, A, A.T, D) == [C, A*A.T, D]
+
+
+def test_determinant():
+    assert det(2*C) == 2**n*det(C)
+    assert det(2*C*D) == 2**n*det(C)*det(D)
+    assert det(3*C*A*A.T*D) == 3**n*det(C)*det(A*A.T)*det(D)
+
+
+def test_doit():
+    assert MatMul(C, 2, D).args == (C, 2, D)
+    assert MatMul(C, 2, D).doit().args == (2, C, D)
+    assert MatMul(C, Transpose(D*C)).args == (C, Transpose(D*C))
+    assert MatMul(C, Transpose(D*C)).doit(deep=True).args == (C, C.T, D.T)
+
+
+def test_doit_drills_down():
+    X = ImmutableMatrix([[1, 2], [3, 4]])
+    Y = ImmutableMatrix([[2, 3], [4, 5]])
+    assert MatMul(X, MatPow(Y, 2)).doit() == X*Y**2
+    assert MatMul(C, Transpose(D*C)).doit().args == (C, C.T, D.T)
+
+
+def test_doit_deep_false_still_canonical():
+    assert (MatMul(C, Transpose(D*C), 2).doit(deep=False).args ==
+            (2, C, Transpose(D*C)))
+
+
+def test_matmul_scalar_Matrix_doit():
+    # Issue 9053
+    X = Matrix([[1, 2], [3, 4]])
+    assert MatMul(2, X).doit() == 2*X
+
+
+def test_matmul_sympify():
+    assert isinstance(MatMul(eye(1), eye(1)).args[0], Basic)
+
+
+def test_collapse_MatrixBase():
+    A = Matrix([[1, 1], [1, 1]])
+    B = Matrix([[1, 2], [3, 4]])
+    assert MatMul(A, B).doit() == ImmutableMatrix([[4, 6], [4, 6]])
+
+
+def test_refine():
+    assert refine(C*C.T*D, Q.orthogonal(C)).doit() == D
+
+    kC = k*C
+    assert refine(kC*C.T, Q.orthogonal(C)).doit() == k*Identity(n)
+    assert refine(kC* kC.T, Q.orthogonal(C)).doit() == (k**2)*Identity(n)
+
+def test_matmul_no_matrices():
+    assert MatMul(1) == 1
+    assert MatMul(n, m) == n*m
+    assert not isinstance(MatMul(n, m), MatMul)
+
+def test_matmul_args_cnc():
+    assert MatMul(n, A, A.T).args_cnc() == [[n], [A, A.T]]
+    assert MatMul(A, A.T).args_cnc() == [[], [A, A.T]]
+
+@XFAIL
+def test_matmul_args_cnc_symbols():
+    # Not currently supported
+    a, b = symbols('a b', commutative=False)
+    assert MatMul(n, a, b, A, A.T).args_cnc() == [[n], [a, b, A, A.T]]
+    assert MatMul(n, a, A, b, A.T).args_cnc() == [[n], [a, A, b, A.T]]
+
+def test_issue_12950():
+    M = Matrix([[Symbol("x")]]) * MatrixSymbol("A", 1, 1)
+    assert MatrixSymbol("A", 1, 1).as_explicit()[0]*Symbol('x') == M.as_explicit()[0]
+
+def test_construction_with_Mul():
+    assert Mul(C, D) == MatMul(C, D)
+    assert Mul(D, C) == MatMul(D, C)
+
+def test_construction_with_mul():
+    assert mul(C, D) == MatMul(C, D)
+    assert mul(D, C) == MatMul(D, C)
+    assert mul(C, D) != MatMul(D, C)
+
+def test_generic_identity():
+    assert MatMul.identity == GenericIdentity()
+    assert MatMul.identity != S.One
+
+
+def test_issue_23519():
+    N = Symbol("N", integer=True)
+    M1 = MatrixSymbol("M1", N, N)
+    M2 = MatrixSymbol("M2", N, N)
+    I = Identity(N)
+    z = (M2 + 2 * (M2 + I) * M1 + I)
+    assert z.coeff(M1) == 2*I + 2*M2
+
+
+def test_shape_error():
+    A = MatrixSymbol('A', 2, 2)
+    B = MatrixSymbol('B', 3, 3)
+    raises(ShapeError, lambda: MatMul(A, B))

llmeval-env/lib/python3.10/site-packages/sympy/matrices/expressions/tests/test_matpow.py

@@ -0,0 +1,217 @@
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.simplify.powsimp import powsimp
+from sympy.testing.pytest import raises
+from sympy.core.expr import unchanged
+from sympy.core import symbols, S
+from sympy.matrices import Identity, MatrixSymbol, ImmutableMatrix, ZeroMatrix, OneMatrix, Matrix
+from sympy.matrices.common import NonSquareMatrixError
+from sympy.matrices.expressions import MatPow, MatAdd, MatMul
+from sympy.matrices.expressions.inverse import Inverse
+from sympy.matrices.expressions.matexpr import MatrixElement
+
+n, m, l, k = symbols('n m l k', integer=True)
+A = MatrixSymbol('A', n, m)
+B = MatrixSymbol('B', m, l)
+C = MatrixSymbol('C', n, n)
+D = MatrixSymbol('D', n, n)
+E = MatrixSymbol('E', m, n)
+
+
+def test_entry_matrix():
+    X = ImmutableMatrix([[1, 2], [3, 4]])
+    assert MatPow(X, 0)[0, 0] == 1
+    assert MatPow(X, 0)[0, 1] == 0
+    assert MatPow(X, 1)[0, 0] == 1
+    assert MatPow(X, 1)[0, 1] == 2
+    assert MatPow(X, 2)[0, 0] == 7
+
+
+def test_entry_symbol():
+    from sympy.concrete import Sum
+    assert MatPow(C, 0)[0, 0] == 1
+    assert MatPow(C, 0)[0, 1] == 0
+    assert MatPow(C, 1)[0, 0] == C[0, 0]
+    assert isinstance(MatPow(C, 2)[0, 0], Sum)
+    assert isinstance(MatPow(C, n)[0, 0], MatrixElement)
+
+
+def test_as_explicit_symbol():
+    X = MatrixSymbol('X', 2, 2)
+    assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2))
+    assert MatPow(X, 1).as_explicit() == X.as_explicit()
+    assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2
+    assert MatPow(X, n).as_explicit() == ImmutableMatrix([
+        [(X ** n)[0, 0], (X ** n)[0, 1]],
+        [(X ** n)[1, 0], (X ** n)[1, 1]],
+    ])
+
+    a = MatrixSymbol("a", 3, 1)
+    b = MatrixSymbol("b", 3, 1)
+    c = MatrixSymbol("c", 3, 1)
+
+    expr = (a.T*b)**S.Half
+    assert expr.as_explicit() == Matrix([[sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])]])
+
+    expr = c*(a.T*b)**S.Half
+    m = sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])
+    assert expr.as_explicit() == Matrix([[c[0, 0]*m], [c[1, 0]*m], [c[2, 0]*m]])
+
+    expr = (a*b.T)**S.Half
+    denom = sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])
+    expected = (a*b.T).as_explicit()/denom
+    assert expr.as_explicit() == expected
+
+    expr = X**-1
+    det = X[0, 0]*X[1, 1] - X[1, 0]*X[0, 1]
+    expected = Matrix([[X[1, 1], -X[0, 1]], [-X[1, 0], X[0, 0]]])/det
+    assert expr.as_explicit() == expected
+
+    expr = X**m
+    assert expr.as_explicit() == X.as_explicit()**m
+
+
+def test_as_explicit_matrix():
+    A = ImmutableMatrix([[1, 2], [3, 4]])
+    assert MatPow(A, 0).as_explicit() == ImmutableMatrix(Identity(2))
+    assert MatPow(A, 1).as_explicit() == A
+    assert MatPow(A, 2).as_explicit() == A**2
+    assert MatPow(A, -1).as_explicit() == A.inv()
+    assert MatPow(A, -2).as_explicit() == (A.inv())**2
+    # less expensive than testing on a 2x2
+    A = ImmutableMatrix([4])
+    assert MatPow(A, S.Half).as_explicit() == A**S.Half
+
+
+def test_doit_symbol():
+    assert MatPow(C, 0).doit() == Identity(n)
+    assert MatPow(C, 1).doit() == C
+    assert MatPow(C, -1).doit() == C.I
+    for r in [2, S.Half, S.Pi, n]:
+        assert MatPow(C, r).doit() == MatPow(C, r)
+
+
+def test_doit_matrix():
+    X = ImmutableMatrix([[1, 2], [3, 4]])
+    assert MatPow(X, 0).doit() == ImmutableMatrix(Identity(2))
+    assert MatPow(X, 1).doit() == X
+    assert MatPow(X, 2).doit() == X**2
+    assert MatPow(X, -1).doit() == X.inv()
+    assert MatPow(X, -2).doit() == (X.inv())**2
+    # less expensive than testing on a 2x2
+    assert MatPow(ImmutableMatrix([4]), S.Half).doit() == ImmutableMatrix([2])
+    X = ImmutableMatrix([[0, 2], [0, 4]]) # det() == 0
+    raises(ValueError, lambda: MatPow(X,-1).doit())
+    raises(ValueError, lambda: MatPow(X,-2).doit())
+
+
+def test_nonsquare():
+    A = MatrixSymbol('A', 2, 3)
+    B = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
+    for r in [-1, 0, 1, 2, S.Half, S.Pi, n]:
+        raises(NonSquareMatrixError, lambda: MatPow(A, r))
+        raises(NonSquareMatrixError, lambda: MatPow(B, r))
+
+
+def test_doit_equals_pow(): #17179
+    X = ImmutableMatrix ([[1,0],[0,1]])
+    assert MatPow(X, n).doit() == X**n == X
+
+
+def test_doit_nested_MatrixExpr():
+    X = ImmutableMatrix([[1, 2], [3, 4]])
+    Y = ImmutableMatrix([[2, 3], [4, 5]])
+    assert MatPow(MatMul(X, Y), 2).doit() == (X*Y)**2
+    assert MatPow(MatAdd(X, Y), 2).doit() == (X + Y)**2
+
+
+def test_identity_power():
+    k = Identity(n)
+    assert MatPow(k, 4).doit() == k
+    assert MatPow(k, n).doit() == k
+    assert MatPow(k, -3).doit() == k
+    assert MatPow(k, 0).doit() == k
+    l = Identity(3)
+    assert MatPow(l, n).doit() == l
+    assert MatPow(l, -1).doit() == l
+    assert MatPow(l, 0).doit() == l
+
+
+def test_zero_power():
+    z1 = ZeroMatrix(n, n)
+    assert MatPow(z1, 3).doit() == z1
+    raises(ValueError, lambda:MatPow(z1, -1).doit())
+    assert MatPow(z1, 0).doit() == Identity(n)
+    assert MatPow(z1, n).doit() == z1
+    raises(ValueError, lambda:MatPow(z1, -2).doit())
+    z2 = ZeroMatrix(4, 4)
+    assert MatPow(z2, n).doit() == z2
+    raises(ValueError, lambda:MatPow(z2, -3).doit())
+    assert MatPow(z2, 2).doit() == z2
+    assert MatPow(z2, 0).doit() == Identity(4)
+    raises(ValueError, lambda:MatPow(z2, -1).doit())
+
+
+def test_OneMatrix_power():
+    o = OneMatrix(3, 3)
+    assert o ** 0 == Identity(3)
+    assert o ** 1 == o
+    assert o * o == o ** 2 == 3 * o
+    assert o * o * o == o ** 3 == 9 * o
+
+    o = OneMatrix(n, n)
+    assert o * o == o ** 2 == n * o
+    # powsimp necessary as n ** (n - 2) * n does not produce n ** (n - 1)
+    assert powsimp(o ** (n - 1) * o) == o ** n == n ** (n - 1) * o
+
+
+def test_transpose_power():
+    from sympy.matrices.expressions.transpose import Transpose as TP
+
+    assert (C*D).T**5 == ((C*D)**5).T == (D.T * C.T)**5
+    assert ((C*D).T**5).T == (C*D)**5
+
+    assert (C.T.I.T)**7 == C**-7
+    assert (C.T**l).T**k == C**(l*k)
+
+    assert ((E.T * A.T)**5).T == (A*E)**5
+    assert ((A*E).T**5).T**7 == (A*E)**35
+    assert TP(TP(C**2 * D**3)**5).doit() == (C**2 * D**3)**5
+
+    assert ((D*C)**-5).T**-5 == ((D*C)**25).T
+    assert (((D*C)**l).T**k).T == (D*C)**(l*k)
+
+
+def test_Inverse():
+    assert Inverse(MatPow(C, 0)).doit() == Identity(n)
+    assert Inverse(MatPow(C, 1)).doit() == Inverse(C)
+    assert Inverse(MatPow(C, 2)).doit() == MatPow(C, -2)
+    assert Inverse(MatPow(C, -1)).doit() == C
+
+    assert MatPow(Inverse(C), 0).doit() == Identity(n)
+    assert MatPow(Inverse(C), 1).doit() == Inverse(C)
+    assert MatPow(Inverse(C), 2).doit() == MatPow(C, -2)
+    assert MatPow(Inverse(C), -1).doit() == C
+
+
+def test_combine_powers():
+    assert (C ** 1) ** 1 == C
+    assert (C ** 2) ** 3 == MatPow(C, 6)
+    assert (C ** -2) ** -3 == MatPow(C, 6)
+    assert (C ** -1) ** -1 == C
+    assert (((C ** 2) ** 3) ** 4) ** 5 == MatPow(C, 120)
+    assert (C ** n) ** n == C ** (n ** 2)
+
+
+def test_unchanged():
+    assert unchanged(MatPow, C, 0)
+    assert unchanged(MatPow, C, 1)
+    assert unchanged(MatPow, Inverse(C), -1)
+    assert unchanged(Inverse, MatPow(C, -1), -1)
+    assert unchanged(MatPow, MatPow(C, -1), -1)
+    assert unchanged(MatPow, MatPow(C, 1), 1)
+
+
+def test_no_exponentiation():
+    # if this passes, Pow.as_numer_denom should recognize
+    # MatAdd as exponent
+    raises(NotImplementedError, lambda: 3**(-2*C))