sage: x.apply_univariate_polynomial(p)
0
+ The characteristic polynomials of the zero and unit elements
+ should be what we think they are in a subalgebra, too::
+
+ sage: J = RealCartesianProductEJA(3)
+ sage: p1 = J.one().characteristic_polynomial()
+ sage: q1 = J.zero().characteristic_polynomial()
+ sage: e0,e1,e2 = J.gens()
+ sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
+ sage: p2 = A.one().characteristic_polynomial()
+ sage: q2 = A.zero().characteristic_polynomial()
+ sage: p1 == p2
+ True
+ sage: q1 == q2
+ True
+
"""
p = self.parent().characteristic_polynomial()
return p(*self.to_vector())
sage: x.is_invertible() == (x.det() != 0)
True
+ Ensure that the determinant is multiplicative on an associative
+ subalgebra as in Faraut and Koranyi's Proposition II.2.2::
+
+ 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).det() == x.det()*y.det()
+ True
+
"""
P = self.parent()
r = P.rank()
"""
B = self.parent().natural_basis()
- W = B[0].matrix_space()
+ W = self.parent().natural_basis_space()
return W.linear_combination(zip(B,self.to_vector()))
+ def norm(self):
+ """
+ The norm of this element with respect to :meth:`inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealCartesianProductEJA)
+
+ EXAMPLES::
+
+ sage: J = RealCartesianProductEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.norm()
+ sqrt(2)
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.norm()
+ 2
+
+ """
+ return self.inner_product(self).sqrt()
+
+
def operator(self):
"""
Return the left-multiplication-by-this-element
Property 2 (multiply on the right for :trac:`28272`):
- sage: alpha = QQ.random_element()
+ sage: alpha = J.base_ring().random_element()
sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
True
sage: set_random_seed()
sage: A = random_eja().zero().subalgebra_generated_by()
sage: A
- Euclidean Jordan algebra of dimension 0 over Rational Field
+ Euclidean Jordan algebra of dimension 0 over...
sage: A.one()
0
sage: x = J.random_element()
sage: y = J.random_element()
sage: z = J.random_element()
- sage: a = QQ.random_element();
+ 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*z.trace_inner_product(y) )
raise TypeError("'other' must live in the same algebra")
return (self*other).trace()
+
+
+ def trace_norm(self):
+ """
+ The norm of this element with respect to :meth:`trace_inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealCartesianProductEJA)
+
+ EXAMPLES::
+
+ sage: J = RealCartesianProductEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.trace_norm()
+ sqrt(2)
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.trace_norm()
+ 2*sqrt(2)
+
+ """
+ return self.trace_inner_product(self).sqrt()