def __new__(cls, *args, **kwargs):
check = kwargs.get('check', True)
if not args:
return GenericIdentity()
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = filter(lambda i: GenericIdentity() != i, args)
args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
factor, matrices = obj.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
# Should it be
#
# return Basic.__neq__(cls, factor, GenericIdentity()) ?
return factor
return obj
def test_generic_identity():
I = GenericIdentity()
A = MatrixSymbol("A", n, n)
assert I == I
assert I != A
assert A != I
assert I.is_Identity
assert I**-1 == I
raises(TypeError, lambda: I.shape)
raises(TypeError, lambda: I.rows)
raises(TypeError, lambda: I.cols)
assert MatMul() == I
assert MatMul(I, A) == MatMul(A)
# Make sure it is hashable
hash(I)
class MatMul(MatrixExpr, Mul):
"""
A product of matrix expressions
Examples
========
>>> from sympy import MatMul, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 4)
>>> B = MatrixSymbol('B', 4, 3)
>>> C = MatrixSymbol('C', 3, 6)
>>> MatMul(A, B, C)
A*B*C
"""
is_MatMul = True
identity = GenericIdentity()
def __new__(cls, *args, **kwargs):
check = kwargs.get('check', True)
if not args:
return cls.identity
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = filter(lambda i: cls.identity != i, args)
args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
factor, matrices = obj.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
# Should it be
#
# return Basic.__neq__(cls, factor, GenericIdentity()) ?
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, **kwargs):
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
def f():
counter = 1
while True:
yield Dummy("i_%i" % counter)
counter += 1
dummy_generator = kwargs.get("dummy_generator", f())
for i in range(1, len(matrices)):
indices[i] = next(dummy_generator)
for i, arg in enumerate(matrices[:-1]):
ind_ranges[i] = arg.shape[1] - 1
matrices = [arg._entry(indices[i], indices[i+1], dummy_generator=dummy_generator) for i, arg in enumerate(matrices)]
expr_in_sum = Mul.fromiter(matrices)
if any(v.has(ImmutableMatrix) for v in matrices):
expand = True
result = coeff*Sum(
expr_in_sum,
*zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
)
# Don't waste time in result.doit() if the sum bounds are symbolic
if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
expand = False
return result.doit() if expand else result
def as_coeff_matrices(self):
scalars = [x for x in self.args if not x.is_Matrix]
matrices = [x for x in self.args if x.is_Matrix]
coeff = Mul(*scalars)
if coeff.is_commutative is False:
raise NotImplementedError("noncommutative scalars in MatMul are not supported.")
return coeff, matrices
def as_coeff_mmul(self):
coeff, matrices = self.as_coeff_matrices()
return coeff, MatMul(*matrices)
def _eval_transpose(self):
"""Transposition of matrix multiplication.
Notes
=====
The following rules are applied.
Transposition for matrix multiplied with another matrix:
`\\left(A B\\right)^{T} = B^{T} A^{T}`
Transposition for matrix multiplied with scalar:
`\\left(c A\\right)^{T} = c A^{T}`
References
==========
.. [1] https://en.wikipedia.org/wiki/Transpose
"""
coeff, matrices = self.as_coeff_matrices()
return MatMul(
coeff, *[transpose(arg) for arg in matrices[::-1]]).doit()
def _eval_adjoint(self):
return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()
def _eval_trace(self):
factor, mmul = self.as_coeff_mmul()
if factor != 1:
from .trace import trace
return factor * trace(mmul.doit())
else:
raise NotImplementedError("Can't simplify any further")
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import Determinant
factor, matrices = self.as_coeff_matrices()
square_matrices = only_squares(*matrices)
return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))
def _eval_inverse(self):
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
# treat scalar*MatrixSymbol or scalar*MatPow separately
expr = canonicalize(MatMul(*args))
return expr
# Needed for partial compatibility with Mul
def args_cnc(self, **kwargs):
coeff_c = [x for x in self.args if x.is_commutative]
coeff_nc = [x for x in self.args if not x.is_commutative]
return [coeff_c, coeff_nc]
def _eval_derivative_matrix_lines(self, x):
from .transpose import Transpose
with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
lines = []
for ind in with_x_ind:
left_args = self.args[:ind]
right_args = self.args[ind+1:]
if right_args:
right_mat = MatMul.fromiter(right_args)
else:
right_mat = Identity(self.shape[1])
if left_args:
left_rev = MatMul.fromiter([Transpose(i).doit() if i.is_Matrix else i for i in reversed(left_args)])
else:
left_rev = Identity(self.shape[0])
d = self.args[ind]._eval_derivative_matrix_lines(x)
for i in d:
i.append_first(left_rev)
i.append_second(right_mat)
lines.append(i)
return lines
def test_generic_identity():
assert MatMul.identity == GenericIdentity()
assert MatMul.identity != S.One
class MatMul(MatrixExpr):
precedence = 45
"""
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_commutative = True
identity = GenericIdentity()
def __new__(cls, *args, **kwargs):
# check = kwargs.get('check', True)
check = kwargs.get('check', False)
if not args:
return cls.identity
if len(args) == 1:
return args[0]
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = list(map(sympify, args))
if any(arg.is_MatMul for arg in args):
def generator():
for arg in args:
if arg.is_MatMul:
yield from arg.args
else:
yield arg
args = [*generator()]
coeffs = []
matrices = []
def append(mat):
if matrices:
last = matrices[-1]
if last.is_MatPow:
if mat.is_MatPow:
if last.base == mat.base:
matrices[-1] = last.func(last.base,
last.exp + mat.exp)
return
elif last.base == mat:
matrices[-1] = last.func(last.base, last.exp + 1)
return
elif last == mat:
if mat._eval_inverse() == last:
matrices.pop()
else:
matrices[-1] = MatPow(last, 2)
return
matrices.append(mat)
for arg in args:
if not arg.is_Mul:
append(arg)
continue
coeff = []
matrix = []
for t in arg.args:
if t.shape:
matrix.append(t)
else:
coeff.append(t)
if coeff:
coeffs.append(Mul(*coeff))
append(Mul(*matrix))
else:
append(arg)
if not matrices:
return Identity(args[0].shape[-1])
matrices = [*filter(lambda X: not X.is_Identity, matrices)]
if len(matrices) == 1:
mat = matrices.pop()
else:
mat = Basic.__new__(cls, *matrices)
factor, matrices = mat.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
# Should it be
#
# return Basic.__neq__(cls, factor, GenericIdentity()) ?
mat = factor
if coeffs:
mat = Mul(*coeffs) * mat
return mat
def argmax_shape(self):
import numpy as np
return np.argmax([len(arg.shape) for arg in self.args])
@property
def shape(self):
dimension = self.args[self.argmax_shape()].shape
dimension = dimension[:-2]
m = self.args[0]
if len(m.shape) == 1:
if len(self.args[1].shape) == 1:
assert m.shape[0] == self.args[1].shape[0]
else:
assert m.shape[0] == self.args[1].shape[
-2], "self.args[0].shape = %s, self.args[1].shape = %s" % (
self.args[0].shape, self.args[1].shape)
else:
assert len(m.shape) >= 2
dimension += (m.shape[-2], )
last_shape = self.args[-1].shape
if len(last_shape) > 1:
dimension += (last_shape[-1], )
return dimension
# matrices = [arg for arg in self.args if arg.is_Matrix]
# return (matrices[0].rows, matrices[-1].cols)
def _entry(self, i, j=None, expand=True, **kwargs):
if j is None:
if len(self.args[0].shape) == 1:
return self.args[0] @ self.func(*self.args[1:])[:, i]
return self.args[0][i] @ self.func(*self.args[1:])
if isinstance(i, slice):
start, stop = i.start, i.stop
if start is None:
if stop is None:
return self.func(*self.args[:-1]) @ self.args[-1][:, j]
start = 0
if stop is None:
stop = self.shape[0]
return
if expand:
from sympy import Dummy, Sum, 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
def f():
counter = 1
while True:
yield Dummy("i_%i" % counter)
counter += 1
dummy_generator = kwargs.get("dummy_generator", f())
for i in range(1, len(matrices)):
indices[i] = next(dummy_generator)
for i, arg in enumerate(matrices[:-1]):
ind_ranges[i] = arg.shape[1] - 1
matrices = [
arg._entry(indices[i],
indices[i + 1],
dummy_generator=dummy_generator)
for i, arg in enumerate(matrices)
]
expr_in_sum = Mul.fromiter(matrices)
if any(v.has(ImmutableMatrix) for v in matrices):
expand = True
result = coeff * Sum(
expr_in_sum,
*zip(indices[1:-1], [0] * len(ind_ranges), ind_ranges))
# Don't waste time in result.doit() if the sum bounds are symbolic
if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
expand = False
return result.doit() if expand else result
else:
return self._entry(i)[:, j]
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]
scalars = [x for x in self.args if not x.shape]
matrices = [x for x in self.args if x.shape]
coeff = Mul(*scalars)
# if coeff.is_commutative == False:
# raise NotImplementedError("noncommutative scalars in MatMul are not supported.")
return coeff, matrices
def as_coeff_mmul(self):
coeff, matrices = self.as_coeff_matrices()
return coeff, MatMul(*matrices)
def _eval_adjoint(self):
return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()
def _eval_trace(self):
factor, mmul = self.as_coeff_mmul()
if factor != 1:
from .trace import trace
return factor * trace(mmul.doit())
else:
raise NotImplementedError("Can't simplify any further")
def _eval_determinant(self):
# from sympy.matrices.expressions.determinant import Det
from sympy import det
factor, matrices = self.as_coeff_matrices()
square_matrices = only_squares(*matrices)
return factor**self.rows * Mul(*list(map(det, square_matrices)))
def _eval_inverse(self):
try:
return MatMul(*[arg.inverse() 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', False)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
# treat scalar*MatrixSymbol or scalar*MatPow separately
expr = canonicalize(MatMul(*args))
return expr
# Needed for partial compatibility with Mul
def args_cnc(self, **kwargs):
coeff_c = [x for x in self.args if x.is_commutative]
coeff_nc = [x for x in self.args if not x.is_commutative]
return [coeff_c, coeff_nc]
def _eval_derivative_matrix_lines(self, x):
from .transpose import Transpose
with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
lines = []
for ind in with_x_ind:
left_args = self.args[:ind]
right_args = self.args[ind + 1:]
if right_args:
right_mat = MatMul.fromiter(right_args)
else:
right_mat = Identity(self.shape[1])
if left_args:
left_rev = MatMul.fromiter([
Transpose(i).doit() if i.is_Matrix else i
for i in reversed(left_args)
])
else:
left_rev = Identity(self.shape[0])
d = self.args[ind]._eval_derivative_matrix_lines(x)
for i in d:
i.append_first(left_rev)
i.append_second(right_mat)
lines.append(i)
return lines
def simplify(self, **_):
this = any_zeros(self)
if this != self:
return this
from sympy import exp
if len(self.args) == 2 and all(
isinstance(arg, exp) for arg in self.args):
if len(self.args[0].shape) < len(self.args[1].shape):
from sympy.concrete import summations
return summations.Sum(
exp(self.args[0].arg + self.args[1].arg.T))
this = self.simplifyProduct()
if this is not self:
return this
return self
def simplifyProduct(self):
from sympy.concrete.products import MatProduct
for i, prod in enumerate(self.args):
if isinstance(prod, MatProduct):
before = self.func(*self.args[:i])
after = self.func(*self.args[i + 1:])
_prod = prod.try_absorb_forward(before)
if _prod:
return _prod @ after
_prod = prod.try_absorb_backward(after)
if _prod:
return before @ _prod
return self
def expand(self, free_symbol=None, deep=True, **_):
if not deep:
return MatrixExpr.expand(self)
from sympy.concrete.expr_with_limits import LAMBDA
from sympy.concrete.summations import Sum
if len(self.args) > 2:
return MatrixExpr.expand(self)
A, B = self.args
if A.is_MatPow:
return self
if A.is_Concatenate:
if B.is_Concatenate and len(A.shape) == 1:
if len(A.args) == len(B.args):
sgm = None
for a, b in zip(A.args, B.args):
if a.shape:
product = a @ b
if product.is_MatMul and len(product.args) == 2:
product = product.expand()
else:
product = a * b
if sgm is None:
sgm = product
else:
sgm += product
return sgm
else:
return self
else:
args = [self.func(arg, B).simplify() for arg in A.args]
if deep:
args = [
arg.expand(deep=True) if arg.is_MatMul else arg
for arg in args
]
return A.func(*args)
if A.is_Transpose:
if B.is_Transpose:
return (B.arg @ A.arg).expand().T
if A.arg.is_Concatenate and B.is_Concatenate:
A_T = A.arg
if len(A_T.args) == len(B.args):
B_T = B._eval_transpose()
if B_T is not None:
# A @ B = A.T.T @ B.T.T = (B.T @ A.T).T
return (B_T @ A_T).expand().T
sgm = None
for a, b in zip(A_T.args, B.args):
if len(a.shape) == 1 and len(b.shape) == 1:
n = a.shape[0]
if b.shape[0] == n:
i = a.generate_free_symbol(b.free_symbols,
integer=True)
j = a.generate_free_symbol(b.free_symbols
| {i},
integer=True)
product = LAMBDA[j:n,
i:n](a[i] * b[j]).simplify()
else:
return self
else:
if not b.shape:
product = a * b
elif a.rows == b.shape[0]:
product = (a.T @ b).simplify()
if product.is_MatMul and len(
product.args) == 2:
X = product.args[1]
if X.is_Transpose and X.arg.is_Concatenate:
product = product.expand()
else:
return self
if sgm is None:
sgm = product
else:
sgm += product
return sgm
return self
if B.is_Concatenate:
return self
if B.is_Transpose and B.arg.is_Concatenate:
return (B.arg @ A.T).expand().T
if A.is_MatProduct:
return self
kwargs = {'free_symbol': free_symbol, 'generator': self}
def generate_k_limit(A, B, excludes=None, **kwargs):
if A.is_LAMBDA or not B.is_LAMBDA:
if excludes:
excludes |= B.free_symbols
else:
excludes = B.free_symbols
return A.generate_int_limit(0, excludes, **kwargs)
if excludes:
excludes |= A.free_symbols
else:
excludes = A.free_symbols
return B.generate_int_limit(0 if len(B.shape) == 1 else 1,
excludes, **kwargs)
if len(A.shape) > 1:
i_limit = A.generate_int_limit(1, **kwargs)
i, *_ = i_limit
if len(B.shape) > 1:
j_limit = B.generate_int_limit(0, {i}, **kwargs)
j, *_ = j_limit
k_limit = generate_k_limit(A, B, {i, j}, **kwargs)
k, *_ = k_limit
assert i != k and k != j and i != j
return LAMBDA(
Sum(A[i, k] * B[k, j], k_limit).simplify(), j_limit,
i_limit).simplify()
else:
k_limit = generate_k_limit(A, B, {i}, **kwargs)
k, *_ = k_limit
assert i != k
return LAMBDA(
Sum(A[i, k] * B[k], k_limit).simplify(),
i_limit).simplify()
else:
# print('A.shape =', A.shape)
if len(B.shape) > 1:
if B.shape[-1].is_Integer:
k_limit = generate_k_limit(A, B, **kwargs)
k, *_ = k_limit
args = []
for j in range(B.shape[-1]):
args.append(Sum(A[k] * B[k, j], k_limit).simplify())
return Concatenate(*args)
else:
# print('B.shape =', B.shape)
j_limit = B.generate_int_limit(0, **kwargs)
j, *_ = j_limit
k_limit = generate_k_limit(A, B, {j}, **kwargs)
k, *_ = k_limit
assert k != j
return LAMBDA(
Sum(A[k] * B[k, j], k_limit).simplify(),
j_limit).simplify()
k_limit = generate_k_limit(A, B, **kwargs)
k, *_ = k_limit
return Sum(A[k] * B[k], k_limit).simplify()
def _eval_is_integer(self):
for elem in self.args:
is_integer = elem.is_integer
if is_integer:
continue
return is_integer
return True
@property
def domain(self):
from sympy import Interval, oo
from sympy.sets.sets import CartesianSpace
shape = self.shape
interval = Interval(-oo, oo, integer=self.is_integer)
if shape:
return CartesianSpace(interval, *shape)
return interval
def _sympystr(self, p):
from sympy.core.mul import _keep_coeff
from sympy.printing.precedence import precedence
c, m = self.as_coeff_mmul()
if c.is_number and c < 0:
expr = _keep_coeff(-c, m)
sign = "-"
level = precedence(expr)
else:
sign = ""
level = precedence(self)
return sign + ' @ '.join(
p.parenthesize(arg, level) for arg in self.args)
def _latex(self, p):
from sympy import MatMul
from sympy.printing.precedence import precedence_traditional
parens = lambda x: p.parenthesize(x, precedence_traditional(self),
False)
args = self.args
args = list(args)
if isinstance(self, MatMul) and self._coeff_isneg():
if args[0] == -1:
args = args[1:]
else:
args[0] = -args[0]
return '- ' + r' \times '.join(map(parens, args))
else:
return r' \times '.join(map(parens, args))
def as_ordered_factors(self, **_):
return [self]
def _eval_is_extended_real(self):
return fuzzy_and(arg.is_extended_real for arg in self.args)
def _eval_is_extended_positive(self):
return fuzzy_and(arg.is_extended_positive for arg in self.args)
def _eval_is_extended_negative(self):
return fuzzy_and(arg.is_extended_negative for arg in self.args)
def _eval_is_finite(self):
return fuzzy_and(arg.is_finite for arg in self.args)
@classmethod
def class_key(cls):
return 3, 0, cls.__name__
@property
def atomic_dtype(self):
dtype = None
for arg in self.args:
_dtype = arg.atomic_dtype
if dtype is None or dtype in _dtype:
dtype = _dtype
return dtype
def domain_defined(self, x):
from sympy import S
if x.atomic_dtype.is_set:
return S.UniversalSet
domain = x.domain
for arg in self.args:
domain &= arg.domain_defined(x)
return domain
def domain_definition(self):
eq = MatrixExpr.domain_definition(self)
for arg in self.args:
eq &= arg.domain_definition()
return eq
def _eval_transpose(self):
"""Transposition of matrix multiplication.
Notes
=====
The following rules are applied.
Transposition for matrix multiplied with another matrix:
`\\left(A B\\right)^{T} = B^{T} A^{T}`
Transposition for matrix multiplied with scalar:
`\\left(c A\\right)^{T} = c A^{T}`
References
==========
.. [1] https://en.wikipedia.org/wiki/Transpose
"""
return self.func(*(arg.T for arg in self.args[::-1]))
def _subs(self, old, new, **hints):
if old.is_MatMul:
args = old.args
for i in range(len(self.args) - len(args) + 1):
if all(self.args[j] == args[j - i]
for j in range(i, i + len(args))):
return self.func(*self.args[:i] +
(new.args if new.is_MatMul else (new, )) +
self.args[i + len(args):]).simplify()
return MatrixExpr._subs(self, old, new, **hints)
def distribute(self):
for i, arg in enumerate(self.args):
if arg.is_Sum or arg.is_Integral:
args = [*self.args]
args[i] = arg.function
function = self.func(*args).powsimp()
return arg.func(function, *arg.limits)
if arg.is_Plus:
args = [*self.args]
if i > 0:
left = arg.func(*(self.func(*args[:i]) @ a
for a in arg.args))
right = args[i + 1:]
if right:
return left @ self.func(*right)
else:
return left
else:
return self
return self