venv/lib/python3.11/site-packages/sympy/matrices/common.py

"""
A module containing deprecated matrix mixin classes.

The classes in this module are deprecated and will be removed in a future
release. They are kept here for backwards compatibility in case downstream
code was subclassing them.

Importing anything else from this module is deprecated so anything here
should either not be used or should be imported from somewhere else.
"""
from __future__ import annotations
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.logic import fuzzy_and, FuzzyBool
from sympy.core.numbers import Integer
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 sympy.utilities.exceptions import sympy_deprecation_warning
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


# These exception types were previously defined in this module but were moved
# to exceptions.py. We reimport them here for backwards compatibility in case
# downstream code was importing them from here.
from .exceptions import ( # noqa: F401
    MatrixError, ShapeError, NonSquareMatrixError, NonInvertibleMatrixError,
    NonPositiveDefiniteMatrixError
)


_DEPRECATED_MIXINS = (
    'MatrixShaping',
    'MatrixSpecial',
    'MatrixProperties',
    'MatrixOperations',
    'MatrixArithmetic',
    'MatrixCommon',
    'MatrixDeterminant',
    'MatrixReductions',
    'MatrixSubspaces',
    'MatrixEigen',
    'MatrixCalculus',
    'MatrixDeprecated',
)


class _MatrixDeprecatedMeta(type):

    #
    # Override the default __instancecheck__ implementation to ensure that
    # e.g. isinstance(M, MatrixCommon) still works when M is one of the
    # matrix classes. Matrix no longer inherits from MatrixCommon so
    # isinstance(M, MatrixCommon) would now return False by default.
    #
    # There were lots of places in the codebase where this was being done
    # so it seems likely that downstream code may be doing it too. All use
    # of these mixins is deprecated though so we give a deprecation warning
    # unconditionally if they are being used with isinstance.
    #
    # Any code seeing this deprecation warning should be changed to use
    # isinstance(M, MatrixBase) instead which also works in previous versions
    # of SymPy.
    #

    def __instancecheck__(cls, instance):

        sympy_deprecation_warning(
            f"""
            Checking whether an object is an instance of {cls.__name__} is
            deprecated.

            Use `isinstance(obj, Matrix)` instead of `isinstance(obj, {cls.__name__})`.
            """,
            deprecated_since_version="1.13",
            active_deprecations_target="deprecated-matrix-mixins",
            stacklevel=3,
        )

        from sympy.matrices.matrixbase import MatrixBase
        from sympy.matrices.matrices import (
            MatrixDeterminant,
            MatrixReductions,
            MatrixSubspaces,
            MatrixEigen,
            MatrixCalculus,
            MatrixDeprecated
        )

        all_mixins = (
            MatrixRequired,
            MatrixShaping,
            MatrixSpecial,
            MatrixProperties,
            MatrixOperations,
            MatrixArithmetic,
            MatrixCommon,
            MatrixDeterminant,
            MatrixReductions,
            MatrixSubspaces,
            MatrixEigen,
            MatrixCalculus,
            MatrixDeprecated
        )

        if cls in all_mixins and isinstance(instance, MatrixBase):
            return True
        else:
            return super().__instancecheck__(instance)


class MatrixRequired(metaclass=_MatrixDeprecatedMeta):
    """Deprecated mixin class for making matrix classes."""

    rows: int
    cols: int
    _simplify = None

    def __init_subclass__(cls, **kwargs):

        # Warn if any downstream code is subclassing this class or any of the
        # deprecated mixin classes that are all ultimately subclasses of this
        # class.
        #
        # We don't want to warn about the deprecated mixins themselves being
        # created, but only about them being used as mixins by downstream code.
        # Otherwise just importing this module would trigger a warning.
        # Ultimately the whole module should be deprecated and removed but for
        # SymPy 1.13 it is premature to do that given that this module was the
        # main way to import matrix exception types in all previous versions.

        if cls.__name__ not in _DEPRECATED_MIXINS:
            sympy_deprecation_warning(
                f"""
                Inheriting from the Matrix mixin classes is deprecated.

                The class {cls.__name__} is subclassing a deprecated mixin.
                """,
                deprecated_since_version="1.13",
                active_deprecations_target="deprecated-matrix-mixins",
                stacklevel=3,
            )

        super().__init_subclass__(**kwargs)

    @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):
            for i in range(1, M.shape[0] + 1):
                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):
                    continue
                sub_blocks.append(M[:i, :i])
                if M.shape != M[:i, :i].shape:
                    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.matrixbase 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:

        >>> print(m.is_anti_symmetric(simplify=False))
        None

        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.matrixbase.MatrixCommon.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.matrixbase.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.matrixbase.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.matrixbase.MatrixBase.cross
        sympy.matrices.matrixbase.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:
            if exp.is_Float:
                exp = Integer(exp)
            # 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: bool = True


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))


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.matrixbase 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__))