Generateurv2/backend/env/lib/python3.10/site-packages/sympy/matrices/expressions/matexpr.py

from typing import Tuple as tTuple

from sympy.core.logic import FuzzyBool

from functools import wraps, reduce
import collections

from sympy.core import S, Symbol, Integer, Basic, Expr, Mul, Add
from sympy.core.decorators import call_highest_priority
from sympy.core.compatibility import SYMPY_INTS, default_sort_key
from sympy.core.symbol import Str
from sympy.core.sympify import SympifyError, _sympify
from sympy.functions import conjugate, adjoint
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.common import NonSquareMatrixError
from sympy.simplify import simplify
from sympy.matrices.matrices import MatrixKind
from sympy.utilities.misc import filldedent
from sympy.multipledispatch import dispatch


def _sympifyit(arg, retval=None):
    # This version of _sympifyit sympifies MutableMatrix objects
    def deco(func):
        @wraps(func)
        def __sympifyit_wrapper(a, b):
            try:
                b = _sympify(b)
                return func(a, b)
            except SympifyError:
                return retval

        return __sympifyit_wrapper

    return deco


class MatrixExpr(Expr):
    """Superclass for Matrix Expressions

    MatrixExprs represent abstract matrices, linear transformations represented
    within a particular basis.

    Examples
    ========

    >>> from sympy import MatrixSymbol
    >>> A = MatrixSymbol('A', 3, 3)
    >>> y = MatrixSymbol('y', 3, 1)
    >>> x = (A.T*A).I * A * y

    See Also
    ========

    MatrixSymbol, MatAdd, MatMul, Transpose, Inverse
    """

    # Should not be considered iterable by the
    # sympy.core.compatibility.iterable function. Subclass that actually are
    # iterable (i.e., explicit matrices) should set this to True.
    _iterable = False

    _op_priority = 11.0

    is_Matrix = True  # type: bool
    is_MatrixExpr = True  # type: bool
    is_Identity = None  # type: FuzzyBool
    is_Inverse = False
    is_Transpose = False
    is_ZeroMatrix = False
    is_MatAdd = False
    is_MatMul = False

    is_commutative = False
    is_number = False
    is_symbol = False
    is_scalar = False

    kind = MatrixKind()

    def __new__(cls, *args, **kwargs):
        args = map(_sympify, args)
        return Basic.__new__(cls, *args, **kwargs)

    # The following is adapted from the core Expr object

    @property
    def shape(self) -> tTuple[Expr, Expr]:
        raise NotImplementedError

    @property
    def _add_handler(self):
        return MatAdd

    @property
    def _mul_handler(self):
        return MatMul

    def __neg__(self):
        return MatMul(S.NegativeOne, self).doit()

    def __abs__(self):
        raise NotImplementedError

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__radd__')
    def __add__(self, other):
        return MatAdd(self, other, check=True).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__add__')
    def __radd__(self, other):
        return MatAdd(other, self, check=True).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rsub__')
    def __sub__(self, other):
        return MatAdd(self, -other, check=True).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__sub__')
    def __rsub__(self, other):
        return MatAdd(other, -self, check=True).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rmul__')
    def __mul__(self, other):
        return MatMul(self, other).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rmul__')
    def __matmul__(self, other):
        return MatMul(self, other).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__mul__')
    def __rmul__(self, other):
        return MatMul(other, self).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__mul__')
    def __rmatmul__(self, other):
        return MatMul(other, self).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rpow__')
    def __pow__(self, other):
        return MatPow(self, other).doit()

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__pow__')
    def __rpow__(self, other):
        raise NotImplementedError("Matrix Power not defined")

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__rtruediv__')
    def __truediv__(self, other):
        return self * other**S.NegativeOne

    @_sympifyit('other', NotImplemented)
    @call_highest_priority('__truediv__')
    def __rtruediv__(self, other):
        raise NotImplementedError()
        #return MatMul(other, Pow(self, S.NegativeOne))

    @property
    def rows(self):
        return self.shape[0]

    @property
    def cols(self):
        return self.shape[1]

    @property
    def is_square(self):
        return self.rows == self.cols

    def _eval_conjugate(self):
        from sympy.matrices.expressions.adjoint import Adjoint
        from sympy.matrices.expressions.transpose import Transpose
        return Adjoint(Transpose(self))

    def as_real_imag(self, deep=True, **hints):
        from sympy import I
        real = S.Half * (self + self._eval_conjugate())
        im = (self - self._eval_conjugate())/(2*I)
        return (real, im)

    def _eval_inverse(self):
        from sympy.matrices.expressions.inverse import Inverse
        return Inverse(self)

    def _eval_transpose(self):
        return Transpose(self)

    def _eval_power(self, exp):
        """
        Override this in sub-classes to implement simplification of powers.  The cases where the exponent
        is -1, 0, 1 are already covered in MatPow.doit(), so implementations can exclude these cases.
        """
        return MatPow(self, exp)

    def _eval_simplify(self, **kwargs):
        if self.is_Atom:
            return self
        else:
            return self.func(*[simplify(x, **kwargs) for x in self.args])

    def _eval_adjoint(self):
        from sympy.matrices.expressions.adjoint import Adjoint
        return Adjoint(self)

    def _eval_derivative_n_times(self, x, n):
        return Basic._eval_derivative_n_times(self, x, n)

    def _eval_derivative(self, x):
        # `x` is a scalar:
        if self.has(x):
            # See if there are other methods using it:
            return super()._eval_derivative(x)
        else:
            return ZeroMatrix(*self.shape)

    @classmethod
    def _check_dim(cls, dim):
        """Helper function to check invalid matrix dimensions"""
        from sympy.core.assumptions import check_assumptions
        ok = check_assumptions(dim, integer=True, nonnegative=True)
        if ok is False:
            raise ValueError(
                "The dimension specification {} should be "
                "a nonnegative integer.".format(dim))


    def _entry(self, i, j, **kwargs):
        raise NotImplementedError(
            "Indexing not implemented for %s" % self.__class__.__name__)

    def adjoint(self):
        return adjoint(self)

    def as_coeff_Mul(self, rational=False):
        """Efficiently extract the coefficient of a product. """
        return S.One, self

    def conjugate(self):
        return conjugate(self)

    def transpose(self):
        from sympy.matrices.expressions.transpose import transpose
        return transpose(self)

    @property
    def T(self):
        '''Matrix transposition'''
        return self.transpose()

    def inverse(self):
        if not self.is_square:
            raise NonSquareMatrixError('Inverse of non-square matrix')
        return self._eval_inverse()

    def inv(self):
        return self.inverse()

    @property
    def I(self):
        return self.inverse()

    def valid_index(self, i, j):
        def is_valid(idx):
            return isinstance(idx, (int, Integer, Symbol, Expr))
        return (is_valid(i) and is_valid(j) and
                (self.rows is None or
                (0 <= i) != False and (i < self.rows) != False) and
                (0 <= j) != False and (j < self.cols) != False)

    def __getitem__(self, key):
        if not isinstance(key, tuple) and isinstance(key, slice):
            from sympy.matrices.expressions.slice import MatrixSlice
            return MatrixSlice(self, key, (0, None, 1))
        if isinstance(key, tuple) and len(key) == 2:
            i, j = key
            if isinstance(i, slice) or isinstance(j, slice):
                from sympy.matrices.expressions.slice import MatrixSlice
                return MatrixSlice(self, i, j)
            i, j = _sympify(i), _sympify(j)
            if self.valid_index(i, j) != False:
                return self._entry(i, j)
            else:
                raise IndexError("Invalid indices (%s, %s)" % (i, j))
        elif isinstance(key, (SYMPY_INTS, Integer)):
            # row-wise decomposition of matrix
            rows, cols = self.shape
            # allow single indexing if number of columns is known
            if not isinstance(cols, Integer):
                raise IndexError(filldedent('''
                    Single indexing is only supported when the number
                    of columns is known.'''))
            key = _sympify(key)
            i = key // cols
            j = key % cols
            if self.valid_index(i, j) != False:
                return self._entry(i, j)
            else:
                raise IndexError("Invalid index %s" % key)
        elif isinstance(key, (Symbol, Expr)):
                raise IndexError(filldedent('''
                    Only integers may be used when addressing the matrix
                    with a single index.'''))
        raise IndexError("Invalid index, wanted %s[i,j]" % self)

    def as_explicit(self):
        """
        Returns a dense Matrix with elements represented explicitly

        Returns an object of type ImmutableDenseMatrix.

        Examples
        ========

        >>> from sympy import Identity
        >>> I = Identity(3)
        >>> I
        I
        >>> I.as_explicit()
        Matrix([
        [1, 0, 0],
        [0, 1, 0],
        [0, 0, 1]])

        See Also
        ========
        as_mutable: returns mutable Matrix type

        """
        if (not isinstance(self.rows, (SYMPY_INTS, Integer))
            or not isinstance(self.cols, (SYMPY_INTS, Integer))):
            raise ValueError(
                'Matrix with symbolic shape '
                'cannot be represented explicitly.')
        from sympy.matrices.immutable import ImmutableDenseMatrix
        return ImmutableDenseMatrix([[self[i, j]
                            for j in range(self.cols)]
                            for i in range(self.rows)])

    def as_mutable(self):
        """
        Returns a dense, mutable matrix with elements represented explicitly

        Examples
        ========

        >>> from sympy import Identity
        >>> I = Identity(3)
        >>> I
        I
        >>> I.shape
        (3, 3)
        >>> I.as_mutable()
        Matrix([
        [1, 0, 0],
        [0, 1, 0],
        [0, 0, 1]])

        See Also
        ========
        as_explicit: returns ImmutableDenseMatrix
        """
        return self.as_explicit().as_mutable()

    def __array__(self):
        from numpy import empty
        a = empty(self.shape, dtype=object)
        for i in range(self.rows):
            for j in range(self.cols):
                a[i, j] = self[i, j]
        return a

    def equals(self, other):
        """
        Test elementwise equality between matrices, potentially of different
        types

        >>> from sympy import Identity, eye
        >>> Identity(3).equals(eye(3))
        True
        """
        return self.as_explicit().equals(other)

    def canonicalize(self):
        return self

    def as_coeff_mmul(self):
        return 1, MatMul(self)

    @staticmethod
    def from_index_summation(expr, first_index=None, last_index=None, dimensions=None):
        r"""
        Parse expression of matrices with explicitly summed indices into a
        matrix expression without indices, if possible.

        This transformation expressed in mathematical notation:

        `\sum_{j=0}^{N-1} A_{i,j} B_{j,k} \Longrightarrow \mathbf{A}\cdot \mathbf{B}`

        Optional parameter ``first_index``: specify which free index to use as
        the index starting the expression.

        Examples
        ========

        >>> from sympy import MatrixSymbol, MatrixExpr, Sum
        >>> from sympy.abc import i, j, k, l, N
        >>> A = MatrixSymbol("A", N, N)
        >>> B = MatrixSymbol("B", N, N)
        >>> expr = Sum(A[i, j]*B[j, k], (j, 0, N-1))
        >>> MatrixExpr.from_index_summation(expr)
        A*B

        Transposition is detected:

        >>> expr = Sum(A[j, i]*B[j, k], (j, 0, N-1))
        >>> MatrixExpr.from_index_summation(expr)
        A.T*B

        Detect the trace:

        >>> expr = Sum(A[i, i], (i, 0, N-1))
        >>> MatrixExpr.from_index_summation(expr)
        Trace(A)

        More complicated expressions:

        >>> expr = Sum(A[i, j]*B[k, j]*A[l, k], (j, 0, N-1), (k, 0, N-1))
        >>> MatrixExpr.from_index_summation(expr)
        A*B.T*A.T
        """
        from sympy import Sum, Mul, Add, MatMul, transpose, trace
        from sympy.strategies.traverse import bottom_up

        def remove_matelement(expr, i1, i2):

            def repl_match(pos):
                def func(x):
                    if not isinstance(x, MatrixElement):
                        return False
                    if x.args[pos] != i1:
                        return False
                    if x.args[3-pos] == 0:
                        if x.args[0].shape[2-pos] == 1:
                            return True
                        else:
                            return False
                    return True
                return func

            expr = expr.replace(repl_match(1),
                lambda x: x.args[0])
            expr = expr.replace(repl_match(2),
                lambda x: transpose(x.args[0]))

            # Make sure that all Mul are transformed to MatMul and that they
            # are flattened:
            rule = bottom_up(lambda x: reduce(lambda a, b: a*b, x.args) if isinstance(x, (Mul, MatMul)) else x)
            return rule(expr)

        def recurse_expr(expr, index_ranges={}):
            if expr.is_Mul:
                nonmatargs = []
                pos_arg = []
                pos_ind = []
                dlinks = {}
                link_ind = []
                counter = 0
                args_ind = []
                for arg in expr.args:
                    retvals = recurse_expr(arg, index_ranges)
                    assert isinstance(retvals, list)
                    if isinstance(retvals, list):
                        for i in retvals:
                            args_ind.append(i)
                    else:
                        args_ind.append(retvals)
                for arg_symbol, arg_indices in args_ind:
                    if arg_indices is None:
                        nonmatargs.append(arg_symbol)
                        continue
                    if isinstance(arg_symbol, MatrixElement):
                        arg_symbol = arg_symbol.args[0]
                    pos_arg.append(arg_symbol)
                    pos_ind.append(arg_indices)
                    link_ind.append([None]*len(arg_indices))
                    for i, ind in enumerate(arg_indices):
                        if ind in dlinks:
                            other_i = dlinks[ind]
                            link_ind[counter][i] = other_i
                            link_ind[other_i[0]][other_i[1]] = (counter, i)
                        dlinks[ind] = (counter, i)
                    counter += 1
                counter2 = 0
                lines = {}
                while counter2 < len(link_ind):
                    for i, e in enumerate(link_ind):
                        if None in e:
                            line_start_index = (i, e.index(None))
                            break
                    cur_ind_pos = line_start_index
                    cur_line = []
                    index1 = pos_ind[cur_ind_pos[0]][cur_ind_pos[1]]
                    while True:
                        d, r = cur_ind_pos
                        if pos_arg[d] != 1:
                            if r % 2 == 1:
                                cur_line.append(transpose(pos_arg[d]))
                            else:
                                cur_line.append(pos_arg[d])
                        next_ind_pos = link_ind[d][1-r]
                        counter2 += 1
                        # Mark as visited, there will be no `None` anymore:
                        link_ind[d] = (-1, -1)
                        if next_ind_pos is None:
                            index2 = pos_ind[d][1-r]
                            lines[(index1, index2)] = cur_line
                            break
                        cur_ind_pos = next_ind_pos
                lines = {k: MatMul.fromiter(v) if len(v) != 1 else v[0] for k, v in lines.items()}
                return [(Mul.fromiter(nonmatargs), None)] + [
                    (MatrixElement(a, i, j), (i, j)) for (i, j), a in lines.items()
                ]
            elif expr.is_Add:
                res = [recurse_expr(i) for i in expr.args]
                d = collections.defaultdict(list)
                for res_addend in res:
                    scalar = 1
                    for elem, indices in res_addend:
                        if indices is None:
                            scalar = elem
                            continue
                        indices = tuple(sorted(indices, key=default_sort_key))
                        d[indices].append(scalar*remove_matelement(elem, *indices))
                        scalar = 1
                return [(MatrixElement(Add.fromiter(v), *k), k) for k, v in d.items()]
            elif isinstance(expr, KroneckerDelta):
                i1, i2 = expr.args
                shape = dimensions
                if shape is None:
                    shape = []
                    for kr_ind in expr.args:
                        if kr_ind not in index_ranges:
                            continue
                        r1, r2 = index_ranges[kr_ind]
                        if r1 != 0:
                            raise ValueError(f"index ranges should start from zero: {index_ranges}")
                        shape.append(r2)
                    if len(shape) == 0:
                        shape = None
                    elif len(shape) == 1:
                        shape = (shape[0] + 1, shape[0] + 1)
                    else:
                        shape = (shape[0] + 1, shape[1] + 1)
                        if shape[0] != shape[1]:
                            raise ValueError(f"upper index ranges should be equal: {index_ranges}")

                identity = Identity(shape[0])
                return [(MatrixElement(identity, i1, i2), (i1, i2))]
            elif isinstance(expr, MatrixElement):
                matrix_symbol, i1, i2 = expr.args
                if i1 in index_ranges:
                    r1, r2 = index_ranges[i1]
                    if r1 != 0 or matrix_symbol.shape[0] != r2+1:
                        raise ValueError("index range mismatch: {} vs. (0, {})".format(
                            (r1, r2), matrix_symbol.shape[0]))
                if i2 in index_ranges:
                    r1, r2 = index_ranges[i2]
                    if r1 != 0 or matrix_symbol.shape[1] != r2+1:
                        raise ValueError("index range mismatch: {} vs. (0, {})".format(
                            (r1, r2), matrix_symbol.shape[1]))
                if (i1 == i2) and (i1 in index_ranges):
                    return [(trace(matrix_symbol), None)]
                return [(MatrixElement(matrix_symbol, i1, i2), (i1, i2))]
            elif isinstance(expr, Sum):
                return recurse_expr(
                    expr.args[0],
                    index_ranges={i[0]: i[1:] for i in expr.args[1:]}
                )
            else:
                return [(expr, None)]

        retvals = recurse_expr(expr)
        factors, indices = zip(*retvals)
        retexpr = Mul.fromiter(factors)
        if len(indices) == 0 or list(set(indices)) == [None]:
            return retexpr
        if first_index is None:
            for i in indices:
                if i is not None:
                    ind0 = i
                    break
            return remove_matelement(retexpr, *ind0)
        else:
            return remove_matelement(retexpr, first_index, last_index)

    def applyfunc(self, func):
        from .applyfunc import ElementwiseApplyFunction
        return ElementwiseApplyFunction(func, self)

@dispatch(MatrixExpr, Expr)
def _eval_is_eq(lhs, rhs): # noqa:F811
    return False

@dispatch(MatrixExpr, MatrixExpr)  # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
    if lhs.shape != rhs.shape:
        return False
    if (lhs - rhs).is_ZeroMatrix:
        return True

def get_postprocessor(cls):
    def _postprocessor(expr):
        # To avoid circular imports, we can't have MatMul/MatAdd on the top level
        mat_class = {Mul: MatMul, Add: MatAdd}[cls]
        nonmatrices = []
        matrices = []
        for term in expr.args:
            if isinstance(term, MatrixExpr):
                matrices.append(term)
            else:
                nonmatrices.append(term)

        if not matrices:
            return cls._from_args(nonmatrices)

        if nonmatrices:
            if cls == Mul:
                for i in range(len(matrices)):
                    if not matrices[i].is_MatrixExpr:
                        # If one of the matrices explicit, absorb the scalar into it
                        # (doit will combine all explicit matrices into one, so it
                        # doesn't matter which)
                        matrices[i] = matrices[i].__mul__(cls._from_args(nonmatrices))
                        nonmatrices = []
                        break

            else:
                # Maintain the ability to create Add(scalar, matrix) without
                # raising an exception. That way different algorithms can
                # replace matrix expressions with non-commutative symbols to
                # manipulate them like non-commutative scalars.
                return cls._from_args(nonmatrices + [mat_class(*matrices).doit(deep=False)])

        if mat_class == MatAdd:
            return mat_class(*matrices).doit(deep=False)
        return mat_class(cls._from_args(nonmatrices), *matrices).doit(deep=False)
    return _postprocessor


Basic._constructor_postprocessor_mapping[MatrixExpr] = {
    "Mul": [get_postprocessor(Mul)],
    "Add": [get_postprocessor(Add)],
}


def _matrix_derivative(expr, x):
    from sympy.tensor.array.array_derivatives import ArrayDerivative
    lines = expr._eval_derivative_matrix_lines(x)

    parts = [i.build() for i in lines]

    from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix

    parts = [[convert_array_to_matrix(j) for j in i] for i in parts]

    def _get_shape(elem):
        if isinstance(elem, MatrixExpr):
            return elem.shape
        return 1, 1

    def get_rank(parts):
        return sum([j not in (1, None) for i in parts for j in _get_shape(i)])

    ranks = [get_rank(i) for i in parts]
    rank = ranks[0]

    def contract_one_dims(parts):
        if len(parts) == 1:
            return parts[0]
        else:
            p1, p2 = parts[:2]
            if p2.is_Matrix:
                p2 = p2.T
            if p1 == Identity(1):
                pbase = p2
            elif p2 == Identity(1):
                pbase = p1
            else:
                pbase = p1*p2
            if len(parts) == 2:
                return pbase
            else:  # len(parts) > 2
                if pbase.is_Matrix:
                    raise ValueError("")
                return pbase*Mul.fromiter(parts[2:])

    if rank <= 2:
        return Add.fromiter([contract_one_dims(i) for i in parts])

    return ArrayDerivative(expr, x)


class MatrixElement(Expr):
    parent = property(lambda self: self.args[0])
    i = property(lambda self: self.args[1])
    j = property(lambda self: self.args[2])
    _diff_wrt = True
    is_symbol = True
    is_commutative = True

    def __new__(cls, name, n, m):
        n, m = map(_sympify, (n, m))
        from sympy import MatrixBase
        if isinstance(name, (MatrixBase,)):
            if n.is_Integer and m.is_Integer:
                return name[n, m]
        if isinstance(name, str):
            name = Symbol(name)
        else:
            name = _sympify(name)
            if not isinstance(name.kind, MatrixKind):
                raise TypeError("First argument of MatrixElement should be a matrix")
        obj = Expr.__new__(cls, name, n, m)
        return obj

    def doit(self, **kwargs):
        deep = kwargs.get('deep', True)
        if deep:
            args = [arg.doit(**kwargs) for arg in self.args]
        else:
            args = self.args
        return args[0][args[1], args[2]]

    @property
    def indices(self):
        return self.args[1:]

    def _eval_derivative(self, v):
        from sympy import Sum, symbols, Dummy

        if not isinstance(v, MatrixElement):
            from sympy import MatrixBase
            if isinstance(self.parent, MatrixBase):
                return self.parent.diff(v)[self.i, self.j]
            return S.Zero

        M = self.args[0]

        m, n = self.parent.shape

        if M == v.args[0]:
            return KroneckerDelta(self.args[1], v.args[1], (0, m-1)) * \
                   KroneckerDelta(self.args[2], v.args[2], (0, n-1))

        if isinstance(M, Inverse):
            i, j = self.args[1:]
            i1, i2 = symbols("z1, z2", cls=Dummy)
            Y = M.args[0]
            r1, r2 = Y.shape
            return -Sum(M[i, i1]*Y[i1, i2].diff(v)*M[i2, j], (i1, 0, r1-1), (i2, 0, r2-1))

        if self.has(v.args[0]):
            return None

        return S.Zero


class MatrixSymbol(MatrixExpr):
    """Symbolic representation of a Matrix object

    Creates a SymPy Symbol to represent a Matrix. This matrix has a shape and
    can be included in Matrix Expressions

    Examples
    ========

    >>> from sympy import MatrixSymbol, Identity
    >>> A = MatrixSymbol('A', 3, 4) # A 3 by 4 Matrix
    >>> B = MatrixSymbol('B', 4, 3) # A 4 by 3 Matrix
    >>> A.shape
    (3, 4)
    >>> 2*A*B + Identity(3)
    I + 2*A*B
    """
    is_commutative = False
    is_symbol = True
    _diff_wrt = True

    def __new__(cls, name, n, m):
        n, m = _sympify(n), _sympify(m)

        cls._check_dim(m)
        cls._check_dim(n)

        if isinstance(name, str):
            name = Str(name)
        obj = Basic.__new__(cls, name, n, m)
        return obj

    @property
    def shape(self):
        return self.args[1], self.args[2]

    @property
    def name(self):
        return self.args[0].name

    def _entry(self, i, j, **kwargs):
        return MatrixElement(self, i, j)

    @property
    def free_symbols(self):
        return {self}

    def _eval_simplify(self, **kwargs):
        return self

    def _eval_derivative(self, x):
        # x is a scalar:
        return ZeroMatrix(self.shape[0], self.shape[1])

    def _eval_derivative_matrix_lines(self, x):
        if self != x:
            first = ZeroMatrix(x.shape[0], self.shape[0]) if self.shape[0] != 1 else S.Zero
            second = ZeroMatrix(x.shape[1], self.shape[1]) if self.shape[1] != 1 else S.Zero
            return [_LeftRightArgs(
                [first, second],
            )]
        else:
            first = Identity(self.shape[0]) if self.shape[0] != 1 else S.One
            second = Identity(self.shape[1]) if self.shape[1] != 1 else S.One
            return [_LeftRightArgs(
                [first, second],
            )]


def matrix_symbols(expr):
    return [sym for sym in expr.free_symbols if sym.is_Matrix]


class _LeftRightArgs:
    r"""
    Helper class to compute matrix derivatives.

    The logic: when an expression is derived by a matrix `X_{mn}`, two lines of
    matrix multiplications are created: the one contracted to `m` (first line),
    and the one contracted to `n` (second line).

    Transposition flips the side by which new matrices are connected to the
    lines.

    The trace connects the end of the two lines.
    """

    def __init__(self, lines, higher=S.One):
        self._lines = [i for i in lines]
        self._first_pointer_parent = self._lines
        self._first_pointer_index = 0
        self._first_line_index = 0
        self._second_pointer_parent = self._lines
        self._second_pointer_index = 1
        self._second_line_index = 1
        self.higher = higher

    @property
    def first_pointer(self):
       return self._first_pointer_parent[self._first_pointer_index]

    @first_pointer.setter
    def first_pointer(self, value):
        self._first_pointer_parent[self._first_pointer_index] = value

    @property
    def second_pointer(self):
        return self._second_pointer_parent[self._second_pointer_index]

    @second_pointer.setter
    def second_pointer(self, value):
        self._second_pointer_parent[self._second_pointer_index] = value

    def __repr__(self):
        built = [self._build(i) for i in self._lines]
        return "_LeftRightArgs(lines=%s, higher=%s)" % (
            built,
            self.higher,
        )

    def transpose(self):
        self._first_pointer_parent, self._second_pointer_parent = self._second_pointer_parent, self._first_pointer_parent
        self._first_pointer_index, self._second_pointer_index = self._second_pointer_index, self._first_pointer_index
        self._first_line_index, self._second_line_index = self._second_line_index, self._first_line_index
        return self

    @staticmethod
    def _build(expr):
        from sympy.core.expr import ExprBuilder
        if isinstance(expr, ExprBuilder):
            return expr.build()
        if isinstance(expr, list):
            if len(expr) == 1:
                return expr[0]
            else:
                return expr[0](*[_LeftRightArgs._build(i) for i in expr[1]])
        else:
            return expr

    def build(self):
        data = [self._build(i) for i in self._lines]
        if self.higher != 1:
            data += [self._build(self.higher)]
        data = [i for i in data]
        return data

    def matrix_form(self):
        if self.first != 1 and self.higher != 1:
            raise ValueError("higher dimensional array cannot be represented")

        def _get_shape(elem):
            if isinstance(elem, MatrixExpr):
                return elem.shape
            return (None, None)

        if _get_shape(self.first)[1] != _get_shape(self.second)[1]:
            # Remove one-dimensional identity matrices:
            # (this is needed by `a.diff(a)` where `a` is a vector)
            if _get_shape(self.second) == (1, 1):
                return self.first*self.second[0, 0]
            if _get_shape(self.first) == (1, 1):
                return self.first[1, 1]*self.second.T
            raise ValueError("incompatible shapes")
        if self.first != 1:
            return self.first*self.second.T
        else:
            return self.higher

    def rank(self):
        """
        Number of dimensions different from trivial (warning: not related to
        matrix rank).
        """
        rank = 0
        if self.first != 1:
            rank += sum([i != 1 for i in self.first.shape])
        if self.second != 1:
            rank += sum([i != 1 for i in self.second.shape])
        if self.higher != 1:
            rank += 2
        return rank

    def _multiply_pointer(self, pointer, other):
        from sympy.core.expr import ExprBuilder
        from ...tensor.array.expressions.array_expressions import ArrayTensorProduct
        from ...tensor.array.expressions.array_expressions import ArrayContraction

        subexpr = ExprBuilder(
            ArrayContraction,
            [
                ExprBuilder(
                    ArrayTensorProduct,
                    [
                        pointer,
                        other
                    ]
                ),
                (1, 2)
            ],
            validator=ArrayContraction._validate
        )

        return subexpr

    def append_first(self, other):
        self.first_pointer *= other

    def append_second(self, other):
        self.second_pointer *= other


def _make_matrix(x):
    from sympy import ImmutableDenseMatrix
    if isinstance(x, MatrixExpr):
        return x
    return ImmutableDenseMatrix([[x]])


from .matmul import MatMul
from .matadd import MatAdd
from .matpow import MatPow
from .transpose import Transpose
from .inverse import Inverse
from .special import ZeroMatrix, Identity