Example 11.11::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinEJA(n)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: while not x.is_invertible():
....: x = J.random_element()
aren't multiples of the identity are regular::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinEJA(n)
+ sage: J = JordanSpinEJA.random_instance()
sage: x = J.random_element()
sage: x == x.coefficient(0)*J.one() or x.degree() == 2
True
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA,
....: random_eja)
TESTS:
The minimal polynomial and the characteristic polynomial coincide
and are known (see Alizadeh, Example 11.11) for all elements of
the spin factor algebra that aren't scalar multiples of the
- identity::
+ identity. We require the dimension of the algebra to be at least
+ two here so that said elements actually exist::
sage: set_random_seed()
- sage: n = ZZ.random_element(2,10)
+ sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
+ sage: n = ZZ.random_element(2, n_max)
sage: J = JordanSpinEJA(n)
sage: y = J.random_element()
sage: while y == y.coefficient(0)*J.one():
sage: x.apply_univariate_polynomial(p)
0
+ The minimal polynomial is invariant under a change of basis,
+ and in particular, a re-scaling of the basis::
+
+ sage: set_random_seed()
+ 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: X = random_matrix(QQ,n)
+ sage: X = X*X.transpose()
+ sage: x1 = J1(X)
+ sage: x2 = J2(X)
+ sage: x1.minimal_polynomial() == x2.minimal_polynomial()
+ True
+
"""
if self.is_zero():
# We would generate a zero-dimensional subalgebra
Alizadeh's Example 11.12::
sage: set_random_seed()
- sage: n = ZZ.random_element(1,10)
- sage: J = JordanSpinEJA(n)
- sage: x = J.random_element()
+ sage: x = JordanSpinEJA.random_instance().random_element()
sage: x_vec = x.to_vector()
+ sage: n = x_vec.degree()
sage: x0 = x_vec[0]
sage: x_bar = x_vec[1:]
sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
TESTS:
- The trace inner product is commutative::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element(); y = J.random_element()
- sage: x.trace_inner_product(y) == y.trace_inner_product(x)
- True
-
- The trace inner product is bilinear::
+ The trace inner product is commutative, bilinear, and satisfies
+ the Jordan axiom:
sage: set_random_seed()
sage: J = random_eja()
- sage: x = J.random_element()
+ sage: x = J.random_element();
sage: y = J.random_element()
sage: z = J.random_element()
+ sage: # commutative
+ sage: x.trace_inner_product(y) == y.trace_inner_product(x)
+ True
+ sage: # bilinear
sage: a = J.base_ring().random_element();
sage: actual = (a*(x+z)).trace_inner_product(y)
sage: expected = ( a*x.trace_inner_product(y) +
....: a*x.trace_inner_product(z) )
sage: actual == expected
True
-
- The trace inner product satisfies the compatibility
- condition in the definition of a Euclidean Jordan algebra::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
+ sage: # jordan axiom
sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
True