from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-
+from mjo.eja.eja_utils import gram_schmidt
class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
"""
1
"""
- def __init__(self, elt):
+ def __init__(self, elt, orthonormalize_basis):
self._superalgebra = elt.parent()
category = self._superalgebra.category().Associative()
V = self._superalgebra.vector_space()
natural_basis=natural_basis)
- # First compute the vector subspace spanned by the powers of
- # the given element.
+ # This list is guaranteed to contain all independent powers,
+ # because it's the maximal set of powers that could possibly
+ # be independent (by a dimension argument).
powers = [ elt**k for k in range(V.dimension()) ]
- power_vectors = [ p.to_vector() for p in powers ]
- # Figure out which powers form a linearly-independent set.
- ind_rows = matrix(field, power_vectors).pivot_rows()
+ if orthonormalize_basis == False:
+ # In this case, we just need to figure out which elements
+ # of the "powers" list are redundant... First compute the
+ # vector subspace spanned by the powers of the given
+ # element.
+ power_vectors = [ p.to_vector() for p in powers ]
- # Pick those out of the list of all powers.
- superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
+ # Figure out which powers form a linearly-independent set.
+ ind_rows = matrix(field, power_vectors).pivot_rows()
- # If our superalgebra is a subalgebra of something else, then
- # these vectors won't have the right coordinates for
- # V.span_of_basis() unless we use V.from_vector() on them.
- basis_vectors = map(power_vectors.__getitem__, ind_rows)
- W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
+ # Pick those out of the list of all powers.
+ superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
- # Now figure out the entries of the right-multiplication
- # matrix for the successive basis elements b0, b1,... of
- # that subspace.
+ # If our superalgebra is a subalgebra of something else, then
+ # these vectors won't have the right coordinates for
+ # V.span_of_basis() unless we use V.from_vector() on them.
+ basis_vectors = map(power_vectors.__getitem__, ind_rows)
+ else:
+ # If we're going to orthonormalize the basis anyway, we
+ # might as well just do Gram-Schmidt on the whole list of
+ # powers. The redundant ones will get zero'd out.
+ superalgebra_basis = gram_schmidt(powers)
+ basis_vectors = [ b.to_vector() for b in superalgebra_basis ]
+
+ W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
n = len(superalgebra_basis)
mult_table = [[W.zero() for i in range(n)] for j in range(n)]
for i in range(n):
sage: J = RealSymmetricEJA(3)
sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
sage: [ K(x^k) for k in range(J.rank()) ]
[f0, f1, f2]
sage: J = RealSymmetricEJA(3)
sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
sage: K.vector_space()
Vector space of degree 6 and dimension 3 over...
User basis matrix: