def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if all(ask(Q.symmetric(arg), assumptions) for arg in mmul.args):
return True
if len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T:
if len(mmul.args) == 2:
return True
return ask(Q.symmetric(MatMul(*mmul.args[1:-1])), assumptions)
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if all(ask(Q.symmetric(arg), assumptions) for arg in mmul.args):
return True
if len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T:
if len(mmul.args) == 2:
return True
return ask(Q.symmetric(MatMul(*mmul.args[1:-1])), assumptions)
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if all(ask(Q.symmetric(arg), assumptions) for arg in mmul.args):
return True
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T:
if len(mmul.args) == 2:
return True
return ask(Q.symmetric(MatMul(*mmul.args[1:-1])), assumptions)
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if all(ask(Q.symmetric(arg), assumptions) for arg in mmul.args):
return True
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T:
if len(mmul.args) == 2:
return True
return ask(Q.symmetric(MatMul(*mmul.args[1:-1])), assumptions)
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if Q.symmetric(expr) in conjuncts(assumptions):
return True
def MatrixSlice(expr, assumptions):
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if not expr.on_diag:
return None
else:
return ask(Q.symmetric(expr.parent), assumptions)
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if Q.symmetric(expr) in conjuncts(assumptions):
return True
def MatrixSlice(expr, assumptions):
# TODO: implement sathandlers system for the matrices.
# Now it duplicates the general fact: Implies(Q.diagonal, Q.symmetric).
if ask(Q.diagonal(expr), assumptions):
return True
if not expr.on_diag:
return None
else:
return ask(Q.symmetric(expr.parent), assumptions)
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.symmetric(base), assumptions)
return None
def MatPow(expr, assumptions):
# only for integer powers
base, exp = expr.args
int_exp = ask(Q.integer(exp), assumptions)
if not int_exp:
return None
non_negative = ask(~Q.negative(exp), assumptions)
if (non_negative or non_negative == False
and ask(Q.invertible(base), assumptions)):
return ask(Q.symmetric(base), assumptions)
return None
def refine_matrixelement(expr, assumptions):
"""
Handler for symmetric part.
Examples
========
>>> from sympy.assumptions.refine import refine_matrixelement
>>> from sympy import MatrixSymbol, Q
>>> X = MatrixSymbol('X', 3, 3)
>>> refine_matrixelement(X[0, 1], Q.symmetric(X))
X[0, 1]
>>> refine_matrixelement(X[1, 0], Q.symmetric(X))
X[0, 1]
"""
from sympy.matrices.expressions.matexpr import MatrixElement
matrix, i, j = expr.args
if ask(Q.symmetric(matrix), assumptions):
if (i - j).could_extract_minus_sign():
return expr
return MatrixElement(matrix, j, i)
def Transpose(expr, assumptions):
return ask(Q.symmetric(expr.arg), assumptions)
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
if Q.symmetric(expr) in conjuncts(assumptions):
return True
def MatAdd(expr, assumptions):
return all(ask(Q.symmetric(arg), assumptions) for arg in expr.args)
def _eval_transpose(self):
if ask(Q.symmetric(self.parent)):
return MatrixSlice(self.parent, self.colslice, self.rowslice)
else:
return super(MatrixSlice, self)._eval_transpose()
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.symmetric(expr.parent), assumptions)
def get_known_facts(x=None):
"""
Facts between unary predicates.
Parameters
==========
x : Symbol, optional
Placeholder symbol for unary facts. Default is ``Symbol('x')``.
Returns
=======
fact : Known facts in conjugated normal form.
"""
if x is None:
x = Symbol('x')
fact = And(
# primitive predicates for extended real exclude each other.
Exclusive(Q.negative_infinite(x), Q.negative(x), Q.zero(x),
Q.positive(x), Q.positive_infinite(x)),
# build complex plane
Exclusive(Q.real(x), Q.imaginary(x)),
Implies(Q.real(x) | Q.imaginary(x), Q.complex(x)),
# other subsets of complex
Exclusive(Q.transcendental(x), Q.algebraic(x)),
Equivalent(Q.real(x),
Q.rational(x) | Q.irrational(x)),
Exclusive(Q.irrational(x), Q.rational(x)),
Implies(Q.rational(x), Q.algebraic(x)),
# integers
Exclusive(Q.even(x), Q.odd(x)),
Implies(Q.integer(x), Q.rational(x)),
Implies(Q.zero(x), Q.even(x)),
Exclusive(Q.composite(x), Q.prime(x)),
Implies(Q.composite(x) | Q.prime(x),
Q.integer(x) & Q.positive(x)),
Implies(Q.even(x) & Q.positive(x) & ~Q.prime(x), Q.composite(x)),
# hermitian and antihermitian
Implies(Q.real(x), Q.hermitian(x)),
Implies(Q.imaginary(x), Q.antihermitian(x)),
Implies(Q.zero(x),
Q.hermitian(x) | Q.antihermitian(x)),
# define finity and infinity, and build extended real line
Exclusive(Q.infinite(x), Q.finite(x)),
Implies(Q.complex(x), Q.finite(x)),
Implies(
Q.negative_infinite(x) | Q.positive_infinite(x), Q.infinite(x)),
# commutativity
Implies(Q.finite(x) | Q.infinite(x), Q.commutative(x)),
# matrices
Implies(Q.orthogonal(x), Q.positive_definite(x)),
Implies(Q.orthogonal(x), Q.unitary(x)),
Implies(Q.unitary(x) & Q.real_elements(x), Q.orthogonal(x)),
Implies(Q.unitary(x), Q.normal(x)),
Implies(Q.unitary(x), Q.invertible(x)),
Implies(Q.normal(x), Q.square(x)),
Implies(Q.diagonal(x), Q.normal(x)),
Implies(Q.positive_definite(x), Q.invertible(x)),
Implies(Q.diagonal(x), Q.upper_triangular(x)),
Implies(Q.diagonal(x), Q.lower_triangular(x)),
Implies(Q.lower_triangular(x), Q.triangular(x)),
Implies(Q.upper_triangular(x), Q.triangular(x)),
Implies(Q.triangular(x),
Q.upper_triangular(x) | Q.lower_triangular(x)),
Implies(Q.upper_triangular(x) & Q.lower_triangular(x), Q.diagonal(x)),
Implies(Q.diagonal(x), Q.symmetric(x)),
Implies(Q.unit_triangular(x), Q.triangular(x)),
Implies(Q.invertible(x), Q.fullrank(x)),
Implies(Q.invertible(x), Q.square(x)),
Implies(Q.symmetric(x), Q.square(x)),
Implies(Q.fullrank(x) & Q.square(x), Q.invertible(x)),
Equivalent(Q.invertible(x), ~Q.singular(x)),
Implies(Q.integer_elements(x), Q.real_elements(x)),
Implies(Q.real_elements(x), Q.complex_elements(x)),
)
return fact
def _(expr, assumptions):
return ask(Q.symmetric(expr.arg), assumptions)
def test_symmetry():
X = MatrixSymbol('x', 10, 10)
Y = X[:5, 5:]
with assuming(Q.symmetric(X)):
assert Y.T == X[5:, :5]
def _eval_transpose(self):
if ask(Q.symmetric(self.parent)):
return MatrixSlice(self.parent, self.colslice, self.rowslice)
else:
return super(MatrixSlice, self)._eval_transpose()
