X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=12166267cd34410e0a03616a4b2ad5c7f5b8c105;hb=6b5e01f0f520475a7de60757a6c70511254ec2c6;hp=07c49473a044411374583b60a15127096c355c2c;hpb=4e1d889fa70b660e789f31adb13f449c9f3a2fb1;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 07c4947..1216626 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -15,9 +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 QuadraticField +from sage.rings.number_field.number_field import NumberField 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.structure.element import is_Matrix from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement @@ -135,13 +136,17 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.zero() natural_basis = self.natural_basis() - if elt not in natural_basis[0].matrix_space(): + basis_space = natural_basis[0].matrix_space() + if elt not in basis_space: raise ValueError("not a naturally-represented algebra element") - # Thanks for nothing! Matrix spaces aren't vector - # spaces in Sage, so we have to figure out its - # natural-basis coordinates ourselves. - V = VectorSpace(elt.base_ring(), elt.nrows()*elt.ncols()) + # Thanks for nothing! Matrix spaces aren't vector spaces in + # Sage, so we have to figure out its natural-basis coordinates + # ourselves. We use the basis space's ring instead of the + # element's ring because the basis space might be an algebraic + # closure whereas the base ring of the 3-by-3 identity matrix + # could be QQ instead of QQbar. + V = VectorSpace(basis_space.base_ring(), elt.nrows()*elt.ncols()) W = V.span_of_basis( _mat2vec(s) for s in natural_basis ) coords = W.coordinate_vector(_mat2vec(elt)) return self.from_vector(coords) @@ -498,8 +503,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Finite family {0: e0, 1: e1, 2: e2} sage: J.natural_basis() ( - [1 0] [0 1] [0 0] - [0 0], [1 0], [0 1] + [1 0] [ 0 1/2*sqrt2] [0 0] + [0 0], [1/2*sqrt2 0], [0 1] ) :: @@ -674,7 +679,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = RealSymmetricEJA(2) sage: J.vector_space() - Vector space of dimension 3 over Rational Field + Vector space of dimension 3 over... """ return self.zero().to_vector().parent().ambient_vector_space() @@ -800,6 +805,19 @@ def random_eja(): def _real_symmetric_basis(n, field): """ Return a basis for the space of real symmetric n-by-n matrices. + + SETUP:: + + sage: from mjo.eja.eja_algebra import _real_symmetric_basis + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: B = _real_symmetric_basis(n, QQbar) + sage: all( M.is_symmetric() for M in B) + True + """ # The basis of symmetric matrices, as matrices, in their R^(n-by-n) # coordinates. @@ -810,8 +828,9 @@ def _real_symmetric_basis(n, field): if i == j: Sij = Eij else: - # Beware, orthogonal but not normalized! Sij = Eij + Eij.transpose() + # Now normalize it. + Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt() S.append(Sij) return tuple(S) @@ -828,11 +847,14 @@ def _complex_hermitian_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) + sage: B = _complex_hermitian_basis(n, QQ) + sage: all( M.is_symmetric() for M in B) True """ - F = QuadraticField(-1, 'I') + R = PolynomialRing(field, 'z') + z = R.gen() + F = NumberField(z**2 + 1, 'I', embedding=CLF(-1).sqrt()) I = F.gen() # This is like the symmetric case, but we need to be careful: @@ -869,7 +891,8 @@ def _quaternion_hermitian_basis(n, field): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) + sage: B = _quaternion_hermitian_basis(n, QQbar) + sage: all( M.is_symmetric() for M in B ) True """ @@ -945,7 +968,9 @@ def _embed_complex_matrix(M): EXAMPLES:: - sage: F = QuadraticField(-1,'i') + sage: R = PolynomialRing(QQ, 'z') + sage: z = R.gen() + sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) sage: x1 = F(4 - 2*i) sage: x2 = F(1 + 2*i) sage: x3 = F(-i) @@ -964,7 +989,9 @@ def _embed_complex_matrix(M): sage: set_random_seed() sage: n = ZZ.random_element(5) - sage: F = QuadraticField(-1, 'i') + sage: R = PolynomialRing(QQ, 'z') + sage: z = R.gen() + sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) sage: X = random_matrix(F, n) sage: Y = random_matrix(F, n) sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y) @@ -1011,7 +1038,9 @@ def _unembed_complex_matrix(M): Unembedding is the inverse of embedding:: sage: set_random_seed() - sage: F = QuadraticField(-1, 'i') + sage: R = PolynomialRing(QQ, 'z') + sage: z = R.gen() + sage: F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) sage: M = random_matrix(F, 3) sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M True @@ -1023,7 +1052,9 @@ def _unembed_complex_matrix(M): if not n.mod(2).is_zero(): raise ValueError("the matrix 'M' must be a complex embedding") - F = QuadraticField(-1, 'i') + R = PolynomialRing(QQ, 'z') + z = R.gen() + F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) i = F.gen() # Go top-left to bottom-right (reading order), converting every @@ -1084,7 +1115,9 @@ def _embed_quaternion_matrix(M): if M.ncols() != n: raise ValueError("the matrix 'M' must be square") - F = QuadraticField(-1, 'i') + R = PolynomialRing(QQ, 'z') + z = R.gen() + F = NumberField(z**2 + 1, 'i', embedding=CLF(-1).sqrt()) i = F.gen() blocks = [] @@ -1171,6 +1204,9 @@ 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): """ @@ -1189,7 +1225,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: e0*e0 e0 sage: e1*e1 - e0 + e2 + 1/2*e0 + 1/2*e2 sage: e2*e2 e2 @@ -1235,8 +1271,33 @@ class RealSymmetricEJA(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(1,5) + sage: J = RealSymmetricEJA(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 = RealSymmetricEJA(n).random_element() + sage: x.operator().matrix().is_symmetric() + True + """ def __init__(self, n, field=QQ, **kwargs): + if n > 1: + # 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') + S = _real_symmetric_basis(n, field) Qs = _multiplication_table_from_matrix_basis(S) @@ -1248,7 +1309,9 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): **kwargs) def inner_product(self, x, y): - return _matrix_ip(x,y) + X = x.natural_representation() + Y = y.natural_representation() + return _real_symmetric_matrix_ip(X,Y) class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):