from sage.matrix.constructor import matrix from sage.misc.cachefunc import cached_method from sage.rings.all import QQ from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra): def __init__(self, elt, orthonormalize_basis): self._superalgebra = elt.parent() category = self._superalgebra.category().Associative() V = self._superalgebra.vector_space() field = self._superalgebra.base_ring() # 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: # Echelonize the matrix ourselves, because otherwise the # call to P.pivot_rows() below can choose a non-optimal # row-reduction algorithm. In particular, scaling can # help over AA because it avoids the RecursionError that # gets thrown when we have to look too hard for a root. # # Beware: QQ supports an entirely different set of "algorithm" # keywords than do AA and RR. algo = None if field is not QQ: algo = "scaled_partial_pivoting" P.echelonize(algorithm=algo) # 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)) 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 ] fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) fdeja.__init__(self._superalgebra, superalgebra_basis, category=category, check_axioms=False) # The rank is the highest possible degree of a minimal # polynomial, and is bounded above by the dimension. We know # in this case that there's an element whose minimal # polynomial has the same degree as the space's dimension # (remember how we constructed the space?), so that must be # its rank too. self.rank.set_cache(self.dimension()) @cached_method 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 superalgebra identity restricted to this algebra is its identity. Note that we can't count on the first basis element being the identity -- it might have been scaled if we orthonormalized the basis. SETUP:: sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES:: sage: J = HadamardEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 sage: x = sum(J.gens()) sage: A = x.subalgebra_generated_by() sage: A.one() f0 sage: A.one().superalgebra_element() e0 + e1 + e2 + e3 + e4 TESTS: The identity element acts like the identity over the rationals:: sage: set_random_seed() sage: x = random_eja(field=QQ).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 identity element acts like the identity over the algebraic reals with an orthonormal basis:: sage: set_random_seed() sage: x = random_eja().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(field=QQ).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().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() return self(self.superalgebra().one())