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)
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 pdf(self, *args):
mu, sigma = Matrix(self.mu), Matrix(self.sigma)
k = len(mu)
args = Matrix(args)
return S(1)/sqrt((2*pi)**(k)*det(sigma))*exp(
-S(1)/2*(mu - args).transpose()*(sigma**(-1)*\
(mu - args)))[0]
def pdf(self, *args):
mu, sigma = self.mu, self.sigma
k = mu.shape[0]
args = ImmutableMatrix(args)
x = args - mu
return (S.One / sqrt((2 * pi)**(k) * det(sigma)) *
exp(Rational(-1, 2) * x.transpose() * (sigma.inv() * x))[0])
def pdf(self, *args):
mu, sigma = self.mu, self.sigma
k = len(mu)
args = ImmutableMatrix(args)
x = args - mu
return S(1)/sqrt((2*pi)**(k)*det(sigma))*exp(
-S(1)/2*x.transpose()*(sigma.inv()*\
x))[0]
def pdf(self, *args):
mu, sigma = self.mu, self.sigma
k = len(mu)
args = ImmutableMatrix(args)
x = args - mu
return S(1)/sqrt((2*pi)**(k)*det(sigma))*exp(
-S(1)/2*x.transpose()*(sigma.inv()*\
x))[0]
def pdf(self, *args):
mu, sigma = self.mu, self.shape_mat
v = S(self.dof)
k = S(len(mu))
sigma_inv = sigma.inv()
args = ImmutableMatrix(args)
x = args - mu
return gamma((k + v)/2)/(gamma(v/2)*(v*pi)**(k/2)*sqrt(det(sigma)))\
*(1 + 1/v*(x.transpose()*sigma_inv*x)[0])**((-v - k)/2)
def pdf(self, *args):
from sympy.functions.special.gamma_functions import gamma
mu, sigma = Matrix(self.mu), Matrix(self.shape_mat)
v = S(self.dof)
k = S(len(mu))
sigma_inv = sigma.inv()
args = Matrix(args)
x = args - mu
return gamma((k + v)/2)/(gamma(v/2)*(v*pi)**(k/2)*sqrt(det(sigma)))\
*(1 + 1/v*(x.transpose()*sigma_inv*x)[0])**((-v - k)/2)
def pdf(self, *args):
from sympy.functions.special.gamma_functions import gamma
mu, sigma = self.mu, self.shape_mat
v = S(self.dof)
k = S(len(mu))
sigma_inv = sigma.inv()
args = ImmutableMatrix(args)
x = args - mu
return gamma((k + v)/2)/(gamma(v/2)*(v*pi)**(k/2)*sqrt(det(sigma)))\
*(1 + 1/v*(x.transpose()*sigma_inv*x)[0])**((-v - k)/2)
def pdf(self, *args):
mu, sigma = self.mu, self.sigma
k = mu.shape[0]
if len(args) == 1 and args[0].is_Matrix:
args = args[0]
else:
args = ImmutableMatrix(args)
x = args - mu
density = S.One / sqrt((2 * pi)**(k) * det(sigma)) * exp(
Rational(-1, 2) * x.transpose() * (sigma.inv() * x))
return MatrixElement(density, 0, 0)
def marginal_distribution(self, indices, sym):
sym = Matrix([Symbol(str(Indexed(sym, i))) for i in indices])
_mu, _sigma = Matrix(self.mu), Matrix(self.sigma)
k = len(self.mu)
for i in range(k):
if i not in indices:
_mu.row_del(i)
_sigma.col_del(i)
_sigma.row_del(i)
return Lambda(sym, S(1)/sqrt((2*pi)**(len(_mu))*det(_sigma))*exp(
-S(1)/2*(_mu - sym).transpose()*(_sigma**(-1)*\
(_mu - sym)))[0])
def _marginal_distribution(self, indices, sym):
sym = ImmutableMatrix([Indexed(sym, i) for i in indices])
_mu, _sigma = self.mu, self.sigma
k = self.mu.shape[0]
for i in range(k):
if i not in indices:
_mu = _mu.row_del(i)
_sigma = _sigma.col_del(i)
_sigma = _sigma.row_del(i)
return Lambda(tuple(sym), S.One/sqrt((2*pi)**(len(_mu))*det(_sigma))*exp(
Rational(-1, 2)*(_mu - sym).transpose()*(_sigma.inv()*\
(_mu - sym)))[0])
def marginal_distribution(self, indices, sym):
sym = ImmutableMatrix([Indexed(sym, i) for i in indices])
_mu, _sigma = self.mu, self.sigma
k = len(self.mu)
for i in range(k):
if i not in indices:
_mu = _mu.row_del(i)
_sigma = _sigma.col_del(i)
_sigma = _sigma.row_del(i)
return Lambda(sym, S(1)/sqrt((2*pi)**(len(_mu))*det(_sigma))*exp(
-S(1)/2*(_mu - sym).transpose()*(_sigma.inv()*\
(_mu - sym)))[0])
def marginal_distribution(self, indices, sym):
sym = ImmutableMatrix([Indexed(sym, i) for i in indices])
_mu, _sigma = self.mu, self.sigma
k = len(self.mu)
for i in range(k):
if i not in indices:
_mu = _mu.row_del(i)
_sigma = _sigma.col_del(i)
_sigma = _sigma.row_del(i)
return Lambda(sym, S(1)/sqrt((2*pi)**(len(_mu))*det(_sigma))*exp(
-S(1)/2*(_mu - sym).transpose()*(_sigma.inv()*\
(_mu - sym)))[0])
def pdf(self, *args):
mu, sigma = self.mu, self.sigma
mu_T = mu.transpose()
k = S(len(mu))
sigma_inv = sigma.inv()
args = ImmutableMatrix(args)
args_T = args.transpose()
x = (mu_T*sigma_inv*mu)[0]
y = (args_T*sigma_inv*args)[0]
v = 1 - k/2
return S(2)/((2*pi)**(S(k)/2)*sqrt(det(sigma)))\
*(y/(2 + x))**(S(v)/2)*besselk(v, sqrt((2 + x)*(y)))\
*exp((args_T*sigma_inv*mu)[0])
def pdf(self, *args):
mu, sigma = self.mu, self.sigma
mu_T = mu.transpose()
k = S(mu.shape[0])
sigma_inv = sigma.inv()
args = ImmutableMatrix(args)
args_T = args.transpose()
x = (mu_T*sigma_inv*mu)[0]
y = (args_T*sigma_inv*args)[0]
v = 1 - k/2
return (2 * (y/(2 + x))**(v/2) * besselk(v, sqrt((2 + x)*y)) *
exp((args_T * sigma_inv * mu)[0]) /
((2 * pi)**(k/2) * sqrt(det(sigma))))
def pdf(self, *args):
from sympy.functions.special.bessel import besselk
mu, sigma = self.mu, self.sigma
mu_T = mu.transpose()
k = S(len(mu))
sigma_inv = sigma.inv()
args = ImmutableMatrix(args)
args_T = args.transpose()
x = (mu_T*sigma_inv*mu)[0]
y = (args_T*sigma_inv*args)[0]
v = 1 - k/2
return S(2)/((2*pi)**(S(k)/2)*sqrt(det(sigma)))\
*(y/(2 + x))**(S(v)/2)*besselk(v, sqrt((2 + x)*(y)))\
*exp((args_T*sigma_inv*mu)[0])
def pdf(self, *args):
from sympy.functions.special.bessel import besselk
mu, sigma = Matrix(self.mu), Matrix(self.sigma)
mu_T = mu.transpose()
k = S(len(mu))
sigma_inv = sigma.inv()
args = Matrix(args)
args_T = args.transpose()
x = (mu_T * sigma_inv * mu)[0]
y = (args_T * sigma_inv * args)[0]
v = 1 - k / 2
return S(2)/((2*pi)**(S(k)/2)*sqrt(det(sigma)))\
*(y/(2 + x))**(S(v)/2)*besselk(v, sqrt((2 + x)*(y)))\
*exp((args_T*sigma_inv*mu)[0])
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 det(self):
from sympy.matrices.expressions.determinant import det
return det(self)
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import det
return det(self.base)**self.exp
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import det
return 1/det(self.arg)
def _eval_determinant(self):
if self._all_square_blocks():
return Mul(*[det(mat) for mat in self.args])
# At least one block is non-square. Since the entire matrix must be square we know there must
# be at least two blocks in this matrix, in which case the entire matrix is necessarily rank-deficient
return S.Zero
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import det
return det(self.arg)
def apply(A):
n = A.shape[0]
k = Symbol.k(integer=True)
return Equality(det(Sum[k:1:n - 1]((Shift(n, 0, n - 1)**k) @ A)),
det(A) * (n - 1) * (-1)**(n - 1))