from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-
class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
"""
SETUP::
sage: actual == expected
True
+ The left-multiplication-by operator for elements in the subalgebra
+ works like it does in the superalgebra, even if we orthonormalize
+ our basis::
+
+ sage: set_random_seed()
+ sage: x = random_eja(AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: y = A.random_element()
+ sage: y.operator()(A.one()) == y
+ True
+
"""
def superalgebra_element(self):
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()
except ValueError:
prefix = prefixen[0]
- if elt.is_zero():
- # Short circuit because 0^0 == 1 is going to make us
- # think we have a one-dimensional algebra otherwise.
- natural_basis = tuple()
- mult_table = tuple()
- rank = 0
- self._vector_space = V.zero_subspace()
- self._superalgebra_basis = []
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra,
- self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
-
-
- # First compute the vector subspace spanned by the powers of
- # the given element.
- superalgebra_basis = [self._superalgebra.one()]
- # If our superalgebra is a subalgebra of something else, then
- # superalgebra.one().to_vector() won't have the right
- # coordinates unless we use V.from_vector() below.
- basis_vectors = [V.from_vector(self._superalgebra.one().to_vector())]
- W = V.span_of_basis(basis_vectors)
- for exponent in range(1, V.dimension()):
- new_power = elt**exponent
- basis_vectors.append( V.from_vector(new_power.to_vector()) )
- try:
- W = V.span_of_basis(basis_vectors)
- superalgebra_basis.append( new_power )
- except ValueError:
- # Vectors weren't independent; bail and keep the
- # last subspace that worked.
- break
-
- # Make the basis hashable for UniqueRepresentation.
- superalgebra_basis = tuple(superalgebra_basis)
-
- # Now figure out the entries of the right-multiplication
- # matrix for the successive basis elements b0, b1,... of
- # that subspace.
+ # 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 ]
+ P = matrix(field, power_vectors)
+
+ 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.
+
+ # Figure out which powers form a linearly-independent set.
+ ind_rows = P.pivot_rows()
+
+ # Pick those out of the list of all powers.
+ superalgebra_basis = tuple(map(powers.__getitem__, ind_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)
+ 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. If this
+ # looks like a roundabout way to orthonormalize, it is.
+ # But converting everything from algebra elements to vectors
+ # to matrices and then back again turns out to be about
+ # as fast as reimplementing our own Gram-Schmidt that
+ # works in an EJA.
+ G,_ = P.gram_schmidt(orthonormal=True)
+ basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
+ superalgebra_basis = [ self._superalgebra.from_vector(b)
+ for b in basis_vectors ]
+
+ 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]
return self.from_vector(coords)
- def one_basis(self):
- """
- Return the basis-element-index of this algebra's unit element.
- """
- return 0
-
def one(self):
"""
Return the multiplicative identity element of this algebra.
The superclass method computes the identity element, which is
- beyond overkill in this case: the algebra identity should be our
- first basis element. We implement this via :meth:`one_basis`
- because that method can optionally be used by other parts of the
- category framework.
+ beyond overkill in this case: the superalgebra identity
+ restricted to this algebra is its identity. Note that we can't
+ count on the first basis element being the identity -- it migth
+ have been scaled if we orthonormalized the basis.
SETUP::
TESTS:
- The identity element acts like the identity::
+ The identity element acts like the identity over the rationals::
sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: x = J.random_element()
- sage: J.one()*x == x and x*J.one() == x
+ sage: x = random_eja().random_element()
+ sage: A = x.subalgebra_generated_by()
+ sage: x = A.random_element()
+ sage: A.one()*x == x and x*A.one() == x
True
- The matrix of the unit element's operator is the identity::
+ The identity element acts like the identity over the algebraic
+ reals with an orthonormal basis::
sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: actual = J.one().operator().matrix()
- sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ sage: x = random_eja(AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: x = A.random_element()
+ sage: A.one()*x == x and x*A.one() == x
+ True
+
+ The matrix of the unit element's operator is the identity over
+ the rationals::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: A = x.subalgebra_generated_by()
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
+
+ The matrix of the unit element's operator is the identity over
+ the algebraic reals with an orthonormal basis::
+
+ sage: set_random_seed()
+ sage: x = random_eja(AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
sage: actual == expected
True
+
"""
if self.dimension() == 0:
return self.zero()
else:
- return self.monomial(self.one_basis())
+ sa_one = self.superalgebra().one().to_vector()
+ sa_coords = self.vector_space().coordinate_vector(sa_one)
+ return self.from_vector(sa_coords)
+
+
+ def natural_basis_space(self):
+ """
+ Return the natural basis space of this algebra, which is identical
+ to that of its superalgebra.
+
+ This is correct "by definition," and avoids a mismatch when the
+ subalgebra is trivial (with no natural basis to infer anything
+ from) and the parent is not.
+ """
+ return self.superalgebra().natural_basis_space()
def superalgebra(self):
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
+ sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over Rational Field
+ Vector space of degree 6 and dimension 3 over...
User basis matrix:
[ 1 0 1 0 0 1]
- [ 0 1 2 3 4 5]
- [10 14 21 19 31 50]
+ [ 1 0 2 0 0 5]
+ [ 1 0 4 0 0 25]
sage: (x^0).to_vector()
(1, 0, 1, 0, 0, 1)
sage: (x^1).to_vector()
- (0, 1, 2, 3, 4, 5)
+ (1, 0, 2, 0, 0, 5)
sage: (x^2).to_vector()
- (10, 14, 21, 19, 31, 50)
+ (1, 0, 4, 0, 0, 25)
"""
return self._vector_space
projects / sage.d.git / blobdiff