# Source code for sympy.matrices.expressions.matmul

from __future__ import print_function, division

from sympy import Number
from sympy.core import Mul, Basic, sympify, Add
from sympy.core.compatibility import range
from sympy.matrices.expressions.transpose import transpose
from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
do_one, new)
from sympy.matrices.expressions.matexpr import (MatrixExpr, ShapeError,
Identity, ZeroMatrix)
from sympy.matrices.matrices import MatrixBase

[docs]class MatMul(MatrixExpr):
"""
A product of matrix expressions

Examples
========

>>> from sympy import MatMul, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 4)
>>> B = MatrixSymbol('B', 4, 3)
>>> C = MatrixSymbol('C', 3, 6)
>>> MatMul(A, B, C)
A*B*C
"""
is_MatMul = True

def __new__(cls, *args, **kwargs):
check = kwargs.get('check', True)

args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
factor, matrices = obj.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
return factor
return obj

@property
def shape(self):
matrices = [arg for arg in self.args if arg.is_Matrix]
return (matrices[0].rows, matrices[-1].cols)

def _entry(self, i, j, expand=True):
from sympy import Dummy, Sum, Mul, ImmutableMatrix, Integer

coeff, matrices = self.as_coeff_matrices()

if len(matrices) == 1:  # situation like 2*X, matmul is just X
return coeff * matrices[0][i, j]

indices = [None]*(len(matrices) + 1)
ind_ranges = [None]*(len(matrices) - 1)
indices[0] = i
indices[-1] = j
for i in range(1, len(matrices)):
indices[i] = Dummy("i_%i" % i)
for i, arg in enumerate(matrices[:-1]):
ind_ranges[i] = arg.shape[1] - 1
matrices = [arg[indices[i], indices[i+1]] for i, arg in enumerate(matrices)]
expr_in_sum = Mul.fromiter(matrices)
if any(v.has(ImmutableMatrix) for v in matrices):
expand = True
result = coeff*Sum(
expr_in_sum,
*zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
)

# Don't waste time in result.doit() if the sum bounds are symbolic
if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
expand = False
return result.doit() if expand else result

def as_coeff_matrices(self):
scalars = [x for x in self.args if not x.is_Matrix]
matrices = [x for x in self.args if x.is_Matrix]
coeff = Mul(*scalars)

return coeff, matrices

def as_coeff_mmul(self):
coeff, matrices = self.as_coeff_matrices()
return coeff, MatMul(*matrices)

def _eval_transpose(self):
return MatMul(*[transpose(arg) for arg in self.args[::-1]]).doit()

return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()

def _eval_trace(self):
factor, mmul = self.as_coeff_mmul()
if factor != 1:
from .trace import trace
return factor * trace(mmul.doit())
else:
raise NotImplementedError("Can't simplify any further")

def _eval_determinant(self):
from sympy.matrices.expressions.determinant import Determinant
factor, matrices = self.as_coeff_matrices()
square_matrices = only_squares(*matrices)
return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))

def _eval_inverse(self):
try:
return MatMul(*[
arg.inverse() if isinstance(arg, MatrixExpr) else arg**-1
for arg in self.args[::-1]]).doit()
except ShapeError:
from sympy.matrices.expressions.inverse import Inverse
return Inverse(self)

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 canonicalize(MatMul(*args))

# Needed for partial compatibility with Mul
def args_cnc(self, **kwargs):
coeff, matrices = self.as_coeff_matrices()
# I don't know how coeff could have noncommutative factors, but this
# handles it.
coeff_c, coeff_nc = coeff.args_cnc(**kwargs)

return coeff_c, coeff_nc + matrices

def validate(*matrices):
""" Checks for valid shapes for args of MatMul """
for i in range(len(matrices)-1):
A, B = matrices[i:i+2]
if A.cols != B.rows:
raise ShapeError("Matrices %s and %s are not aligned"%(A, B))

# Rules

def newmul(*args):
if args[0] == 1:
args = args[1:]
return new(MatMul, *args)

def any_zeros(mul):
if any([arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
for arg in mul.args]):
matrices = [arg for arg in mul.args if arg.is_Matrix]
return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
return mul

def merge_explicit(matmul):
""" Merge explicit MatrixBase arguments

>>> from sympy import MatrixSymbol, eye, Matrix, MatMul, pprint
>>> from sympy.matrices.expressions.matmul import merge_explicit
>>> A = MatrixSymbol('A', 2, 2)
>>> B = Matrix([[1, 1], [1, 1]])
>>> C = Matrix([[1, 2], [3, 4]])
>>> X = MatMul(A, B, C)
>>> pprint(X)
[1  1] [1  2]
A*[    ]*[    ]
[1  1] [3  4]
>>> pprint(merge_explicit(X))
[4  6]
A*[    ]
[4  6]

>>> X = MatMul(B, A, C)
>>> pprint(X)
[1  1]   [1  2]
[    ]*A*[    ]
[1  1]   [3  4]
>>> pprint(merge_explicit(X))
[1  1]   [1  2]
[    ]*A*[    ]
[1  1]   [3  4]
"""
if not any(isinstance(arg, MatrixBase) for arg in matmul.args):
return matmul
newargs = []
last = matmul.args[0]
for arg in matmul.args[1:]:
if isinstance(arg, (MatrixBase, Number)) and isinstance(last, (MatrixBase, Number)):
last = last * arg
else:
newargs.append(last)
last = arg
newargs.append(last)

return MatMul(*newargs)

def xxinv(mul):
""" Y * X * X.I -> Y """
factor, matrices = mul.as_coeff_matrices()
for i, (X, Y) in enumerate(zip(matrices[:-1], matrices[1:])):
try:
if X.is_square and Y.is_square and X == Y.inverse():
I = Identity(X.rows)
return newmul(factor, *(matrices[:i] + [I] + matrices[i+2:]))
except ValueError:  # Y might not be invertible
pass

return mul

def remove_ids(mul):
""" Remove Identities from a MatMul

This is a modified version of sympy.strategies.rm_id.
This is necesssary because MatMul may contain both MatrixExprs and Exprs
as args.

--------
sympy.strategies.rm_id
"""
# Separate Exprs from MatrixExprs in args
factor, mmul = mul.as_coeff_mmul()
# Apply standard rm_id for MatMuls
result = rm_id(lambda x: x.is_Identity is True)(mmul)
if result != mmul:
return newmul(factor, *result.args)  # Recombine and return
else:
return mul

def factor_in_front(mul):
factor, matrices = mul.as_coeff_matrices()
if factor != 1:
return newmul(factor, *matrices)
return mul

rules = (any_zeros, remove_ids, xxinv, unpack, rm_id(lambda x: x == 1),
merge_explicit, factor_in_front, flatten)

canonicalize = exhaust(typed({MatMul: do_one(*rules)}))

def only_squares(*matrices):
""" factor matrices only if they are square """
if matrices[0].rows != matrices[-1].cols:
raise RuntimeError("Invalid matrices being multiplied")
out = []
start = 0
for i, M in enumerate(matrices):
if M.cols == matrices[start].rows:
out.append(MatMul(*matrices[start:i+1]).doit())
start = i+1
return out

from sympy.assumptions.refine import handlers_dict

def refine_MatMul(expr, assumptions):
"""
>>> from sympy import MatrixSymbol, Q, assuming, refine
>>> X = MatrixSymbol('X', 2, 2)
>>> expr = X * X.T
>>> print(expr)
X*X.T
>>> with assuming(Q.orthogonal(X)):
...     print(refine(expr))
I
"""
newargs = []
exprargs = []

for args in expr.args:
if args.is_Matrix:
exprargs.append(args)
else:
newargs.append(args)

last = exprargs[0]
for arg in exprargs[1:]:
if arg == last.T and ask(Q.orthogonal(arg), assumptions):
last = Identity(arg.shape[0])
elif arg == last.conjugate() and ask(Q.unitary(arg), assumptions):
last = Identity(arg.shape[0])
else:
newargs.append(last)
last = arg
newargs.append(last)

return MatMul(*newargs)

handlers_dict['MatMul'] = refine_MatMul