def _(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if (all(ask(Q.positive_definite(arg), assumptions) for arg in mmul.args)
and factor > 0):
return True
if (len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T
and ask(Q.fullrank(mmul.args[0]), assumptions)):
return ask(Q.positive_definite(MatMul(*mmul.args[1:-1])), assumptions)
def MatMul(expr, assumptions):
factor, mmul = expr.as_coeff_mmul()
if (all(ask(Q.positive_definite(arg), assumptions)
for arg in mmul.args) and factor > 0):
return True
if len(mmul.args) >= 2 and mmul.args[0] == mmul.args[-1].T:
return ask(Q.positive_definite(
MatMul(*mmul.args[1:-1])), assumptions)
def MatrixSlice(expr, assumptions):
if not expr.on_diag:
return None
else:
return ask(Q.positive_definite(expr.parent), assumptions)
def Transpose(expr, assumptions):
return ask(Q.positive_definite(expr.arg), assumptions)
def MatrixSymbol(expr, assumptions):
if not expr.is_square:
return False
if Q.positive_definite(expr) in conjuncts(assumptions):
return True
def MatAdd(expr, assumptions):
if all(ask(Q.positive_definite(arg), assumptions)
for arg in expr.args):
return True
def MatrixElement(expr, assumptions):
if (expr.i == expr.j
and ask(Q.positive_definite(expr.parent), assumptions)):
return True
def Determinant(expr, assumptions):
if ask(Q.positive_definite(expr.arg), assumptions):
return True
def Trace(expr, assumptions):
if ask(Q.positive_definite(expr.arg), assumptions):
return True
def MatrixElement(expr, assumptions):
if expr.i == expr.j and ask(Q.positive_definite(expr.parent),
assumptions):
return True
def Determinant(expr, assumptions):
if ask(Q.positive_definite(expr.arg), assumptions):
return True
def Trace(expr, assumptions):
if ask(Q.positive_definite(expr.arg), assumptions):
return True
def MatPow(expr, assumptions):
# a power of a positive definite matrix is positive definite
if ask(Q.positive_definite(expr.args[0]), assumptions):
return True
def MatPow(expr, assumptions):
# a power of a positive definite matrix is positive definite
if ask(Q.positive_definite(expr.args[0]), assumptions):
return True
def _(expr, assumptions):
if (expr.i == expr.j
and ask(Q.positive_definite(expr.parent), assumptions)):
return True
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
