def apply(given):
assert given.is_Unequality
A_det, zero = given.args
assert A_det.is_Determinant
assert zero.is_zero
A = A_det.arg
return Equality(Cofactors(A).T / Determinant(A), Inverse(A), given=given)
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
A = MultivariateNormal('A', [1, 2], [[1, 0], [0, 1]])
assert m == A
assert density(m)(1, 2) == 1/(2*pi)
assert m.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
raises (ValueError, lambda:m[2])
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
raises(ValueError, lambda: marginal_distribution(m))
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N = Normal('N', [1, 2], [[x, 0], [0, y]])
assert density(N)(0, 0) == exp(-((4*x + y)/(2*x*y)))/(2*pi*sqrt(x*y))
raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
# symbolic
n = symbols('n', integer=True, positive=True)
mu = MatrixSymbol('mu', n, 1)
sigma = MatrixSymbol('sigma', n, n)
X = Normal('X', mu, sigma)
assert density(X) == MultivariateNormalDistribution(mu, sigma)
raises (NotImplementedError, lambda: median(m))
# Below tests should work after issue #17267 is resolved
# assert E(X) == mu
# assert variance(X) == sigma
# test symbolic multivariate normal densities
n = 3
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X = density(X)
eval_a = density_X(obs).subs({Sg: eye(3),
mu: Matrix([0, 0, 0]), obs: Matrix([0, 0, 0])}).doit()
eval_b = density_X(0, 0, 0).subs({Sg: eye(3), mu: Matrix([0, 0, 0])}).doit()
assert eval_a == sqrt(2)/(4*pi**Rational(3/2))
assert eval_b == sqrt(2)/(4*pi**Rational(3/2))
n = symbols('n', integer=True, positive=True)
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X_at_obs = density(X)(obs)
expected_density = MatrixElement(
exp((S(1)/2) * (mu.T - obs.T) * Sg**(-1) * (-mu + obs)) / \
sqrt((2*pi)**n * Determinant(Sg)), 0, 0)
assert density_X_at_obs == expected_density
def _eval_determinant(self):
if self.blockshape == (2, 2):
[[A, B], [C, D]] = self.blocks.tolist()
if ask(Q.invertible(A)):
return det(A) * det(D - C * A.I * B)
elif ask(Q.invertible(D)):
return det(D) * det(A - B * D.I * C)
return Determinant(self)
def _eval_determinant(self):
condition = self._is_1x1()
if condition == True:
return S.One
elif condition == False:
return S.Zero
else:
from sympy.matrices.expressions.determinant import Determinant
return Determinant(self)
def prove(Eq):
n = Symbol.n(integer=True)
A = Symbol.A(real=True, shape=(n, n))
a = Symbol.a(real=True, shape=(n, ))
b = Symbol.b(real=True, shape=(n, ))
Eq << apply(Unequal(Determinant(A), 0), Equality(a @ A, b))
Eq << Eq[1] @ Cofactors(A).T
Eq << algebre.matrix.determinant.adjugate.apply(A)
Eq << Eq[-2].subs(Eq[-1])
Eq << algebre.scalar.inequality.equality.apply(Eq[0], Eq[-1])
Eq << algebre.matrix.inequality.determinant.apply(Eq[0]) * Determinant(A)
Eq << Eq[-2].subs(Eq[-1])
def prove(Eq):
n = Symbol.n(integer=True)
A = Symbol.A(real=True, shape=(n, n), given=True)
a = Symbol.a(real=True, shape=(n, ), given=True)
b = Symbol.b(real=True, shape=(n, ), given=True)
Eq << apply(Unequal(Determinant(A), 0), Unequal(a @ A, b))
Eq << ~Eq[-1]
Eq << Eq[-1].split()
Eq << Eq[-2] @ A
Eq <<= Eq[1].subs(Eq[-1])
Eq <<= Eq[-2] & Eq[0]
def prove(Eq):
n = Symbol.n(integer=True, positive=True)
A = Symbol.A(complex=True, shape=(n, n))
Eq << apply(Unequal(Determinant(A), 0))
Eq << algebre.matrix.determinant.adjugate.apply(A)
Eq << algebre.scalar.inequality.equality.apply(Eq[0], Eq[-1])
Eq << Eq[-1].inverse()
Eq << algebre.scalar.inequality.equality.apply(Eq[0], Eq[-1])
Eq << Eq[-1].lhs.args[0].base @ Eq[-1]
Eq << Eq[-1].reversed
def test_MatrixSymbol_determinant():
A = MatrixSymbol('A', 4, 4)
assert A.as_explicit().det() == A[0, 0]*A[1, 1]*A[2, 2]*A[3, 3] - \
A[0, 0]*A[1, 1]*A[2, 3]*A[3, 2] - A[0, 0]*A[1, 2]*A[2, 1]*A[3, 3] + \
A[0, 0]*A[1, 2]*A[2, 3]*A[3, 1] + A[0, 0]*A[1, 3]*A[2, 1]*A[3, 2] - \
A[0, 0]*A[1, 3]*A[2, 2]*A[3, 1] - A[0, 1]*A[1, 0]*A[2, 2]*A[3, 3] + \
A[0, 1]*A[1, 0]*A[2, 3]*A[3, 2] + A[0, 1]*A[1, 2]*A[2, 0]*A[3, 3] - \
A[0, 1]*A[1, 2]*A[2, 3]*A[3, 0] - A[0, 1]*A[1, 3]*A[2, 0]*A[3, 2] + \
A[0, 1]*A[1, 3]*A[2, 2]*A[3, 0] + A[0, 2]*A[1, 0]*A[2, 1]*A[3, 3] - \
A[0, 2]*A[1, 0]*A[2, 3]*A[3, 1] - A[0, 2]*A[1, 1]*A[2, 0]*A[3, 3] + \
A[0, 2]*A[1, 1]*A[2, 3]*A[3, 0] + A[0, 2]*A[1, 3]*A[2, 0]*A[3, 1] - \
A[0, 2]*A[1, 3]*A[2, 1]*A[3, 0] - A[0, 3]*A[1, 0]*A[2, 1]*A[3, 2] + \
A[0, 3]*A[1, 0]*A[2, 2]*A[3, 1] + A[0, 3]*A[1, 1]*A[2, 0]*A[3, 2] - \
A[0, 3]*A[1, 1]*A[2, 2]*A[3, 0] - A[0, 3]*A[1, 2]*A[2, 0]*A[3, 1] + \
A[0, 3]*A[1, 2]*A[2, 1]*A[3, 0]
B = MatrixSymbol('B', 4, 4)
assert Determinant(A + B).doit() == det(A + B) == (A + B).det()
def test_matrix_derivative_with_inverse():
# Cookbook example 61:
expr = a.T * Inverse(X) * b
assert expr.diff(X) == -Inverse(X).T * a * b.T * Inverse(X).T
# Cookbook example 62:
expr = Determinant(Inverse(X))
# Not implemented yet:
# assert expr.diff(X) == -Determinant(X.inv())*(X.inv()).T
# Cookbook example 63:
expr = Trace(A * Inverse(X) * B)
assert expr.diff(X) == -(X**(-1) * B * A * X**(-1)).T
# Cookbook example 64:
expr = Trace(Inverse(X + A))
assert expr.diff(X) == -(Inverse(X + A)).T**2
def test_print_Determinant():
assert mpp.doprint(Determinant(Matrix([[1, 2], [3, 4]]))) == '<mrow><mfenced close="|" open="|"><mfenced close="]" open="["><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></mfenced></mrow>'
def apply(A):
n = A.shape[0]
return Equality(A @ Cofactors(A).T, Determinant(A) * Identity(n))
