X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=a7e56187550111a928823d536dc3ff88347b7424;hb=0848d2345f4813e9f176efef6864d65b1915c495;hp=f7983c857d8fdceffac3dbdfeaa6bf70fd7e811e;hpb=97d4dcfae8b0c5ab1577da2ba3629e0a4169c789;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index f7983c8..a7e5618 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -15,7 +15,7 @@ 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, RLF @@ -802,7 +802,7 @@ def random_eja(): -def _real_symmetric_basis(n, field): +def _real_symmetric_basis(n, field, normalize): """ Return a basis for the space of real symmetric n-by-n matrices. @@ -814,7 +814,7 @@ 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, QQbar, False) sage: all( M.is_symmetric() for M in B) True @@ -829,13 +829,13 @@ 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() + if normalize: + Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt() S.append(Sij) return tuple(S) -def _complex_hermitian_basis(n, field): +def _complex_hermitian_basis(n, field, normalize): """ Returns a basis for the space of complex Hermitian n-by-n matrices. @@ -853,10 +853,8 @@ def _complex_hermitian_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: R = PolynomialRing(QQ, 'z') - sage: z = R.gen() - sage: field = NumberField(z**2 - 2, 'sqrt2', embedding=RLF(2).sqrt()) - sage: B = _complex_hermitian_basis(n, field) + sage: field = QuadraticField(2, 'sqrt2') + sage: B = _complex_hermitian_basis(n, field, False) sage: all( M.is_symmetric() for M in B) True @@ -886,15 +884,17 @@ def _complex_hermitian_basis(n, field): Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose()) S.append(Sij_imag) - # Normalize these with our inner product before handing them back. - # And since we embedded them, we can drop back to the "field" that - # we started with instead of the complex extension "F". - return tuple( (s / _complex_hermitian_matrix_ip(s,s).sqrt()).change_ring(field) - for s in S ) + # Since we embedded these, we can drop back to the "field" that we + # started with instead of the complex extension "F". + S = [ s.change_ring(field) for s in S ] + if normalize: + S = [ s / _complex_hermitian_matrix_ip(s,s).sqrt() for s in S ] + + return tuple(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. @@ -912,7 +912,7 @@ def _quaternion_hermitian_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = _quaternion_hermitian_basis(n, QQ) + sage: B = _quaternion_hermitian_basis(n, QQ, False) sage: all( M.is_symmetric() for M in B ) True @@ -989,9 +989,7 @@ 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) @@ -1010,9 +1008,7 @@ 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) @@ -1059,9 +1055,7 @@ 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 @@ -1137,9 +1131,7 @@ 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 = [] @@ -1317,8 +1309,8 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): True """ - def __init__(self, n, field=QQ, **kwargs): - if n > 1: + def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): + 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. @@ -1328,7 +1320,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): if p.is_irreducible(): field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) - S = _real_symmetric_basis(n, field) + S = _real_symmetric_basis(n, field, normalize_basis) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(RealSymmetricEJA, self) @@ -1415,8 +1407,8 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): True """ - def __init__(self, n, field=QQ, **kwargs): - if n > 1: + def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): + 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. @@ -1426,7 +1418,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): if p.is_irreducible(): field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt()) - S = _complex_hermitian_basis(n, field) + S = _complex_hermitian_basis(n, field, normalize_basis) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(ComplexHermitianEJA, self) @@ -1497,8 +1489,8 @@ 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)