- def subalgebra_generated_by(self):
+ def subalgebra_generated_by(self, orthonormalize_basis=False):
"""
Return the associative subalgebra of the parent EJA generated
by this element.
0
"""
- return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
+ return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
def subalgebra_idempotent(self):
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):
from sage.modules.free_module_element import vector
+from sage.rings.number_field.number_field import NumberField
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.rings.real_lazy import RLF
def _mat2vec(m):
return vector(m.base_ring(), m.list())
+
+def gram_schmidt(v):
+ """
+ Perform Gram-Schmidt on the list ``v`` which are assumed to be
+ vectors over the same base ring. Returns a list of orthonormalized
+ vectors over the smallest extention ring containing the necessary
+ roots.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_utils import gram_schmidt
+
+ EXAMPLES::
+
+ sage: v1 = vector(QQ,(1,2,3))
+ sage: v2 = vector(QQ,(1,-1,6))
+ sage: v3 = vector(QQ,(2,1,-1))
+ sage: v = [v1,v2,v3]
+ sage: u = gram_schmidt(v)
+ sage: [ u_i.inner_product(u_i).sqrt() == 1 for u_i in u ]
+ True
+ sage: u[0].inner_product(u[1]) == 0
+ True
+ sage: u[0].inner_product(u[2]) == 0
+ True
+ sage: u[1].inner_product(u[2]) == 0
+ True
+
+ TESTS:
+
+ Ensure that zero vectors don't get in the way::
+
+ sage: v1 = vector(QQ,(1,2,3))
+ sage: v2 = vector(QQ,(1,-1,6))
+ sage: v3 = vector(QQ,(0,0,0))
+ sage: v = [v1,v2,v3]
+ sage: len(gram_schmidt(v)) == 2
+ True
+
+ """
+ def proj(x,y):
+ return (y.inner_product(x)/x.inner_product(x))*x
+
+ v = list(v) # make a copy, don't clobber the input
+
+ # Drop all zero vectors before we start.
+ v = [ v_i for v_i in v if not v_i.is_zero() ]
+
+ if len(v) == 0:
+ # cool
+ return v
+
+ R = v[0].base_ring()
+
+ # First orthogonalize...
+ for i in xrange(1,len(v)):
+ # Earlier vectors can be made into zero so we have to ignore them.
+ v[i] -= sum( proj(v[j],v[i]) for j in range(i) if not v[j].is_zero() )
+
+ # And now drop all zero vectors again if they were "orthogonalized out."
+ v = [ v_i for v_i in v if not v_i.is_zero() ]
+
+ # Now pretend to normalize, building a new ring R that contains
+ # all of the necessary square roots.
+ norms_squared = [0]*len(v)
+
+ for i in xrange(len(v)):
+ norms_squared[i] = v[i].inner_product(v[i])
+ ns = [norms_squared[i].numerator(), norms_squared[i].denominator()]
+
+ # Do the numerator and denominator separately so that we
+ # adjoin e.g. sqrt(2) and sqrt(3) instead of sqrt(2/3).
+ for j in xrange(len(ns)):
+ PR = PolynomialRing(R, 'z')
+ z = PR.gen()
+ p = z**2 - ns[j]
+ if p.is_irreducible():
+ R = NumberField(p,
+ 'sqrt' + str(ns[j]),
+ embedding=RLF(ns[j]).sqrt())
+
+ # When we're done, we have to change every element's ring to the
+ # extension that we wound up with, and then normalize it (which
+ # should work, since "R" contains its norm now).
+ for i in xrange(len(v)):
+ v[i] = v[i].change_ring(R) / R(norms_squared[i]).sqrt()
+
+ return v
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