from sage.modules.free_module import VectorSpace
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
-# TODO: make this unnecessary somehow.
-from sage.misc.lazy_import import lazy_import
-lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
-lazy_import('mjo.eja.eja_element_subalgebra',
- 'FiniteDimensionalEuclideanJordanElementSubalgebra')
from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
from mjo.eja.eja_utils import _mat2vec
if not self.is_invertible():
raise ValueError("element is not invertible")
+ if self.parent()._charpoly_coefficients.is_in_cache():
+ # We can invert using our charpoly if it will be fast to
+ # compute. If the coefficients are cached, our rank had
+ # better be too!
+ r = self.parent().rank()
+ a = self.characteristic_polynomial().coefficients(sparse=False)
+ return (-1)**(r+1)*sum(a[i+1]*self**i for i in range(r))/self.det()
+
return (~self.quadratic_representation())(self)
whether or not the paren't algebra's zero element is a root
of this element's minimal polynomial.
+ That is... unless the coefficients of our algebra's
+ "characteristic polynomial of" function are already cached!
+ In that case, we just use the determinant (which will be fast
+ as a result).
+
Beware that we can't use the superclass method, because it
relies on the algebra being associative.
else:
return False
+ if self.parent()._charpoly_coefficients.is_in_cache():
+ # The determinant will be quicker than computing the minimal
+ # polynomial from scratch, most likely.
+ return (not self.det().is_zero())
+
# In fact, we only need to know if the constant term is non-zero,
# so we can pass in the field's zero element instead.
zero = self.base_ring().zero()
- def natural_representation(self):
+ def to_matrix(self):
"""
- Return a more-natural representation of this element.
+ Return an (often more natural) representation of this element as a
+ matrix.
- Every finite-dimensional Euclidean Jordan Algebra is a
- direct sum of five simple algebras, four of which comprise
- Hermitian matrices. This method returns the original
- "natural" representation of this element as a Hermitian
- matrix, if it has one. If not, you get the usual representation.
+ Every finite-dimensional Euclidean Jordan Algebra is a direct
+ sum of five simple algebras, four of which comprise Hermitian
+ matrices. This method returns a "natural" matrix
+ representation of this element as either a Hermitian matrix or
+ column vector.
SETUP::
sage: J = ComplexHermitianEJA(3)
sage: J.one()
e0 + e3 + e8
- sage: J.one().natural_representation()
+ sage: J.one().to_matrix()
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
sage: J = QuaternionHermitianEJA(3)
sage: J.one()
e0 + e5 + e14
- sage: J.one().natural_representation()
+ sage: J.one().to_matrix()
[1 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 1]
-
"""
- B = self.parent().natural_basis()
- W = self.parent().natural_basis_space()
- return W.linear_combination(zip(B,self.to_vector()))
+ B = self.parent().matrix_basis()
+ W = self.parent().matrix_space()
+
+ # This is just a manual "from_vector()", but of course
+ # matrix spaces aren't vector spaces in sage, so they
+ # don't have a from_vector() method.
+ return W.linear_combination( zip(B, self.to_vector()) )
def norm(self):
True
"""
+ from mjo.eja.eja_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)