+from itertools import izip
+
from sage.matrix.constructor import matrix
from sage.modules.free_module import VectorSpace
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
elif n == 1:
return self
else:
- return (self.operator()**(n-1))(self)
+ return (self**(n-1))*self
def apply_univariate_polynomial(self, p):
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: x.inner_product(y) in RR
+ sage: x,y = J.random_elements(2)
+ sage: x.inner_product(y) in RLF
True
"""
Test Lemma 1 from Chapter III of Koecher::
sage: set_random_seed()
- sage: J = random_eja()
- sage: u = J.random_element()
- sage: v = J.random_element()
+ sage: u,v = random_eja().random_elements(2)
sage: lhs = u.operator_commutes_with(u*v)
sage: rhs = v.operator_commutes_with(u^2)
sage: lhs == rhs
Chapter III, or from Baes (2.3)::
sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = random_eja().random_elements(2)
sage: Lx = x.operator()
sage: Ly = y.operator()
sage: Lxx = (x*x).operator()
Baes (2.4)::
sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: x,y,z = random_eja().random_elements(3)
sage: Lx = x.operator()
sage: Ly = y.operator()
sage: Lz = z.operator()
Baes (2.5)::
sage: set_random_seed()
- sage: J = random_eja()
- sage: u = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: u,y,z = random_eja().random_elements(3)
sage: Lu = u.operator()
sage: Ly = y.operator()
sage: Lz = z.operator()
sage: set_random_seed()
sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = J.random_elements(2)
sage: (x*y).det() == x.det()*y.det()
True
sage: n_max = RealSymmetricEJA._max_test_case_size()
sage: n = ZZ.random_element(1, n_max)
sage: J1 = RealSymmetricEJA(n,QQ)
- sage: J2 = RealSymmetricEJA(n,QQ,False)
+ sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
sage: X = random_matrix(QQ,n)
sage: X = X*X.transpose()
sage: x1 = J1(X)
"""
B = self.parent().natural_basis()
W = self.parent().natural_basis_space()
- return W.linear_combination(zip(B,self.to_vector()))
+ return W.linear_combination(izip(B,self.to_vector()))
def norm(self):
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = J.random_elements(2)
sage: x.operator()(y) == x*y
True
sage: y.operator()(x) == x*y
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
+ sage: x,y = J.random_elements(2)
sage: Lx = x.operator()
sage: Lxx = (x*x).operator()
sage: Qx = x.quadratic_representation()
sage: not x.is_invertible() or (
....: x.quadratic_representation(x.inverse())*Qx
....: ==
- ....: 2*x.operator()*Qex - Qx )
+ ....: 2*Lx*Qex - Qx )
True
- sage: 2*x.operator()*Qex - Qx == Lxx
+ sage: 2*Lx*Qex - Qx == Lxx
True
Property 5:
- def subalgebra_generated_by(self):
+ def subalgebra_generated_by(self, orthonormalize_basis=False):
"""
Return the associative subalgebra of the parent EJA generated
by this element.
sage: set_random_seed()
sage: x0 = random_eja().random_element()
sage: A = x0.subalgebra_generated_by()
- sage: x = A.random_element()
- sage: y = A.random_element()
- sage: z = A.random_element()
+ sage: x,y,z = A.random_elements(3)
sage: (x*y)*z == x*(y*z)
True
0
"""
- return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
+ return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
def subalgebra_idempotent(self):
sage: set_random_seed()
sage: J = random_eja()
- sage: J.random_element().trace() in J.base_ring()
+ sage: J.random_element().trace() in RLF
True
"""
TESTS:
- The trace inner product is commutative, bilinear, and satisfies
- the Jordan axiom:
+ The trace inner product is commutative, bilinear, and associative::
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element();
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: x,y,z = J.random_elements(3)
sage: # commutative
sage: x.trace_inner_product(y) == y.trace_inner_product(x)
True
....: a*x.trace_inner_product(z) )
sage: actual == expected
True
- sage: # jordan axiom
+ sage: # associative
sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
True