def MatMul(expr, assumptions): factor, mmul = expr.as_coeff_mmul() if all(ask(Q.invertible(arg), assumptions) for arg in mmul.args): return True if any(ask(Q.invertible(arg), assumptions) is False for arg in mmul.args): return False
def MatMul(expr, assumptions): factor, mmul = expr.as_coeff_mmul() if all(ask(Q.invertible(arg), assumptions) for arg in mmul.args): return True if any( ask(Q.invertible(arg), assumptions) is False for arg in mmul.args): return False
def BlockMatrix(expr, assumptions): from sympy.matrices.expressions.blockmatrix import reblock_2x2 if not expr.is_square: return False if expr.blockshape == (1, 1): return ask(Q.invertible(expr.blocks[0, 0]), assumptions) expr = reblock_2x2(expr) if expr.blockshape == (2, 2): [[A, B], [C, D]] = expr.blocks.tolist() if ask(Q.invertible(A), assumptions) == True: invertible = ask(Q.invertible(D - C * A.I * B), assumptions) if invertible is not None: return invertible if ask(Q.invertible(B), assumptions) == True: invertible = ask(Q.invertible(C - D * B.I * A), assumptions) if invertible is not None: return invertible if ask(Q.invertible(C), assumptions) == True: invertible = ask(Q.invertible(B - A * C.I * D), assumptions) if invertible is not None: return invertible if ask(Q.invertible(D), assumptions) == True: invertible = ask(Q.invertible(A - B * D.I * C), assumptions) if invertible is not None: return invertible return None
def _(expr, assumptions): factor, mmul = expr.as_coeff_mmul() if (all(ask(Q.unitary(arg), assumptions) for arg in mmul.args) and abs(factor) == 1): return True if any(ask(Q.invertible(arg), assumptions) is False for arg in mmul.args): return False
def _(expr, assumptions): factor, mmul = expr.as_coeff_mmul() if (all(ask(Q.orthogonal(arg), assumptions) for arg in mmul.args) and factor == 1): return True if any(ask(Q.invertible(arg), assumptions) is False for arg in mmul.args): return False
def MatMul(expr, assumptions): factor, mmul = expr.as_coeff_mmul() if (all(ask(Q.orthogonal(arg), assumptions) for arg in mmul.args) and factor == 1): return True if any(ask(Q.invertible(arg), assumptions) == False for arg in mmul.args): return False
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 if exp.is_negative == False: return ask(Q.invertible(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.complex_elements(base), assumptions) return None
def _(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.diagonal(base), assumptions) return None
def MatrixSlice(expr, assumptions): if not expr.on_diag: return None else: return ask(Q.invertible(expr.parent), assumptions)
def Transpose(expr, assumptions): return ask(Q.invertible(expr.arg), assumptions)
def MatrixSymbol(expr, assumptions): if not expr.is_square: return False if Q.invertible(expr) in conjuncts(assumptions): return True
def _(expr, assumptions): return ask(Q.invertible(expr.arg), assumptions)
def MatrixSymbol(expr, assumptions): if (not expr.is_square or ask(Q.invertible(expr), assumptions) is False): return False if Q.unitary(expr) in conjuncts(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
def _(expr, assumptions): if (not expr.is_square or ask(Q.invertible(expr), assumptions) is False): return False if Q.orthogonal(expr) in conjuncts(assumptions): return True
def BlockDiagMatrix(expr, assumptions): if expr.rowblocksizes != expr.colblocksizes: return None return fuzzy_and( [ask(Q.invertible(a), assumptions) for a in expr.diag])