-"""
+r"""
Representations and constructions for Euclidean Jordan algebras.
A Euclidean Jordan algebra is a Jordan algebra that has some
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
True
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: y = A.random_element()
sage: A.one()*y == y and y*A.one() == y
True
sage: actual == expected
True
sage: x = J.random_element()
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: actual = A.one().operator().matrix()
sage: expected = matrix.identity(A.base_ring(), A.dimension())
sage: actual == expected
# Bypass the hijinks if they won't benefit us.
return super()._charpoly_coefficients()
- # Do the computation over the rationals. The answer will be
- # the same, because all we've done is a change of basis.
- # Then, change back from QQ to our real base ring
+ # Do the computation over the rationals.
a = ( a_i.change_ring(self.base_ring())
for a_i in self.rational_algebra()._charpoly_coefficients() )
- # Otherwise, convert the coordinate variables back to the
- # deorthonormalized ones.
+ # Convert our coordinate variables into deorthonormalized ones
+ # and substitute them into the deorthonormalized charpoly
+ # coefficients.
R = self.coordinate_polynomial_ring()
from sage.modules.free_module_element import vector
X = vector(R, R.gens())
...
TypeError: Illegal initializer for algebraic number
- This causes the following error when we try to scale a matrix of
- complex numbers by an inexact real number::
-
- sage: ComplexHermitianEJA(2,field=RR)
- Traceback (most recent call last):
- ...
- TypeError: Unable to coerce entries (=(1.00000000000000,
- -0.000000000000000)) to coefficients in Algebraic Real Field
-
TESTS:
The dimension of this algebra is `n^2`::