X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=d0e7b074ce41a00f66d6f6dcd487653f3a8b1674;hb=11e681d6320f0b7ddbb834931845b6f4a745da93;hp=12166267cd34410e0a03616a4b2ad5c7f5b8c105;hpb=6b5e01f0f520475a7de60757a6c70511254ec2c6;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 1216626..d0e7b07 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -15,10 +15,10 @@ from sage.misc.prandom import choice from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace from sage.rings.integer_ring import ZZ -from sage.rings.number_field.number_field import NumberField +from sage.rings.number_field.number_field import NumberField, QuadraticField from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.rational_field import QQ -from sage.rings.real_lazy import CLF +from sage.rings.real_lazy import CLF, RLF from sage.structure.element import is_Matrix from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement @@ -416,9 +416,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - if (not x in self) or (not y in self): - raise TypeError("arguments must live in this algebra") - return x.trace_inner_product(y) + X = x.natural_representation() + Y = y.natural_representation() + return self.__class__.natural_inner_product(X,Y) def is_trivial(self): @@ -537,6 +537,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self._natural_basis[0].matrix_space() + @staticmethod + def natural_inner_product(X,Y): + """ + Compute the inner product of two naturally-represented elements. + + For example in the real symmetric matrix EJA, this will compute + the trace inner-product of two n-by-n symmetric matrices. The + default should work for the real cartesian product EJA, the + Jordan spin EJA, and the real symmetric matrices. The others + will have to be overridden. + """ + return (X.conjugate_transpose()*Y).trace() + + @cached_method def one(self): """ @@ -748,7 +762,30 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): return fdeja.__init__(field, mult_table, rank=n, **kwargs) def inner_product(self, x, y): - return _usual_ip(x,y) + """ + Faster to reimplement than to use natural representations. + + SETUP:: + + sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = RealCartesianProductEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y) + True + + """ + return x.to_vector().inner_product(y.to_vector()) def random_eja(): @@ -814,7 +851,7 @@ def _real_symmetric_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = _real_symmetric_basis(n, QQbar) + sage: B = _real_symmetric_basis(n, QQ) sage: all( M.is_symmetric() for M in B) True @@ -829,8 +866,6 @@ def _real_symmetric_basis(n, field): Sij = Eij else: Sij = Eij + Eij.transpose() - # Now normalize it. - Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt() S.append(Sij) return tuple(S) @@ -839,6 +874,12 @@ def _complex_hermitian_basis(n, field): """ Returns a basis for the space of complex Hermitian n-by-n matrices. + Why do we embed these? Basically, because all of numerical linear + algebra assumes that you're working with vectors consisting of `n` + entries from a field and scalars from the same field. There's no way + to tell SageMath that (for example) the vectors contain complex + numbers, while the scalar field is real. + SETUP:: sage: from mjo.eja.eja_algebra import _complex_hermitian_basis @@ -847,7 +888,8 @@ def _complex_hermitian_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = _complex_hermitian_basis(n, QQ) + sage: field = QuadraticField(2, 'sqrt2') + sage: B = _complex_hermitian_basis(n, field) sage: all( M.is_symmetric() for M in B) True @@ -865,24 +907,33 @@ def _complex_hermitian_basis(n, field): S = [] for i in xrange(n): for j in xrange(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) + Eij = matrix(F, n, lambda k,l: k==i and l==j) if i == j: Sij = _embed_complex_matrix(Eij) S.append(Sij) else: - # Beware, orthogonal but not normalized! The second one - # has a minus because it's conjugated. + # The second one has a minus because it's conjugated. Sij_real = _embed_complex_matrix(Eij + Eij.transpose()) S.append(Sij_real) Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) S.append(Sij_imag) - return tuple(S) + + # Since we embedded these, we can drop back to the "field" that we + # started with instead of the complex extension "F". + return tuple( s.change_ring(field) for s in S ) + -def _quaternion_hermitian_basis(n, field): +def _quaternion_hermitian_basis(n, field, normalize): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. + Why do we embed these? Basically, because all of numerical linear + algebra assumes that you're working with vectors consisting of `n` + entries from a field and scalars from the same field. There's no way + to tell SageMath that (for example) the vectors contain complex + numbers, while the scalar field is real. + SETUP:: sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis @@ -891,7 +942,7 @@ def _quaternion_hermitian_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = _quaternion_hermitian_basis(n, QQbar) + sage: B = _quaternion_hermitian_basis(n, QQ, False) sage: all( M.is_symmetric() for M in B ) True @@ -968,9 +1019,7 @@ def _embed_complex_matrix(M): EXAMPLES:: - sage: R = PolynomialRing(QQ, 'z') - sage: z = R.gen() - sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + sage: F = QuadraticField(-1, 'i') sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) @@ -989,9 +1038,7 @@ def _embed_complex_matrix(M): sage: set_random_seed() sage: n = ZZ.random_element(5) - sage: R = PolynomialRing(QQ, 'z') - sage: z = R.gen() - sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + sage: F = QuadraticField(-1, 'i') sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) @@ -1006,8 +1053,8 @@ def _embed_complex_matrix(M): field = M.base_ring() blocks = [] for z in M.list(): - a = z.real() - b = z.imag() + a = z.vector()[0] # real part, I guess + b = z.vector()[1] # imag part, I guess blocks.append(matrix(field, 2, [[a,b],[-b,a]])) # We can drop the imaginaries here. @@ -1038,9 +1085,7 @@ def _unembed_complex_matrix(M): Unembedding is the inverse of embedding:: sage: set_random_seed() - sage: R = PolynomialRing(QQ, 'z') - sage: z = R.gen() - sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + sage: F = QuadraticField(-1, 'i') sage: M = random_matrix(F, 3) sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M True @@ -1052,9 +1097,10 @@ def _unembed_complex_matrix(M): if not n.mod(2).is_zero(): raise ValueError("the matrix 'M' must be a complex embedding") - R = PolynomialRing(QQ, 'z') + field = M.base_ring() # This should already have sqrt2 + R = PolynomialRing(field, 'z') z = R.gen() - F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + F = NumberField(z**2 + 1,'i', embedding=CLF(-1).sqrt()) i = F.gen() # Go top-left to bottom-right (reading order), converting every @@ -1115,9 +1161,7 @@ def _embed_quaternion_matrix(M): if M.ncols() != n: raise ValueError("the matrix 'M' must be square") - R = PolynomialRing(QQ, 'z') - z = R.gen() - F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) + F = QuadraticField(-1, 'i') i = F.gen() blocks = [] @@ -1192,10 +1236,6 @@ def _unembed_quaternion_matrix(M): return matrix(Q, n/4, elements) -# The usual inner product on R^n. -def _usual_ip(x,y): - return x.to_vector().inner_product(y.to_vector()) - # The inner product used for the real symmetric simple EJA. # We keep it as a separate function because e.g. the complex # algebra uses the same inner product, except divided by 2. @@ -1204,9 +1244,6 @@ def _matrix_ip(X,Y): Y_mat = Y.natural_representation() return (X_mat*Y_mat).trace() -def _real_symmetric_matrix_ip(X,Y): - return (X*Y).trace() - class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ @@ -1289,16 +1326,24 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): True """ - def __init__(self, n, field=QQ, **kwargs): - if n > 1: + def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): + S = _real_symmetric_basis(n, field) + + if n > 1 and normalize_basis: # We'll need sqrt(2) to normalize the basis, and this # winds up in the multiplication table, so the whole # algebra needs to be over the field extension. R = PolynomialRing(field, 'z') z = R.gen() - field = NumberField(z**2 - 2, 'sqrt2') + p = z**2 - 2 + if p.is_irreducible(): + field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) + S = [ s.change_ring(field) for s in S ] + self._basis_denormalizers = tuple( + self.__class__.natural_inner_product(s,s).sqrt() + for s in S ) + S = tuple( s/c for (s,c) in zip(S,self._basis_denormalizers) ) - S = _real_symmetric_basis(n, field) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(RealSymmetricEJA, self) @@ -1308,10 +1353,6 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): natural_basis=S, **kwargs) - def inner_product(self, x, y): - X = x.natural_representation() - Y = y.natural_representation() - return _real_symmetric_matrix_ip(X,Y) class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): @@ -1367,9 +1408,42 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: (x*y).inner_product(z) == y.inner_product(x*z) True + Our basis is normalized with respect to the natural inner product:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,4) + sage: J = ComplexHermitianEJA(n) + sage: all( b.norm() == 1 for b in J.gens() ) + True + + Left-multiplication operators are symmetric because they satisfy + the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: x = ComplexHermitianEJA(n).random_element() + sage: x.operator().matrix().is_symmetric() + True + """ - def __init__(self, n, field=QQ, **kwargs): + def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): S = _complex_hermitian_basis(n, field) + + if n > 1 and normalize_basis: + # We'll need sqrt(2) to normalize the basis, and this + # winds up in the multiplication table, so the whole + # algebra needs to be over the field extension. + R = PolynomialRing(field, 'z') + z = R.gen() + p = z**2 - 2 + if p.is_irreducible(): + field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) + S = [ s.change_ring(field) for s in S ] + self._basis_denormalizers = tuple( + self.__class__.natural_inner_product(s,s).sqrt() + for s in S ) + S = tuple( s/c for (s,c) in zip(S,self._basis_denormalizers) ) + Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(ComplexHermitianEJA, self) @@ -1380,16 +1454,12 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): **kwargs) - def inner_product(self, x, y): - # Since a+bi on the diagonal is represented as - # - # a + bi = [ a b ] - # [ -b a ], - # - # we'll double-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/2 - + @staticmethod + def natural_inner_product(X,Y): + Xu = _unembed_complex_matrix(X) + Yu = _unembed_complex_matrix(Y) + # The trace need not be real; consider Xu = (i*I) and Yu = I. + return ((Xu*Yu).trace()).vector()[0] # real part, I guess class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ @@ -1445,8 +1515,8 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): True """ - def __init__(self, n, field=QQ, **kwargs): - S = _quaternion_hermitian_basis(n, field) + def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): + S = _quaternion_hermitian_basis(n, field, normalize_basis) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(QuaternionHermitianEJA, self) @@ -1542,4 +1612,27 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) def inner_product(self, x, y): - return _usual_ip(x,y) + """ + Faster to reimplement than to use natural representations. + + SETUP:: + + sage: from mjo.eja.eja_algebra import JordanSpinEJA + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J = JordanSpinEJA(n) + sage: x = J.random_element() + sage: y = J.random_element() + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: x.inner_product(y) == J.__class__.natural_inner_product(X,Y) + True + + """ + return x.to_vector().inner_product(y.to_vector())