X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=d90d3f2dcb5f4adc5cd6135ac4151ef63c3166c8;hb=6c983c4d14a02b4eff37fd2a07ae6b32b93e611c;hp=adf9581c541bf81e02c612d281a4d671c3b9df74;hpb=9bcc10b6362a5390735bbbf4ef8351b150847359;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index adf9581..d90d3f2 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -60,6 +60,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): self._rank = rank self._natural_basis = natural_basis + # TODO: HACK for the charpoly.. needs redesign badly. + self._basis_normalizers = None + if category is None: category = MagmaticAlgebras(field).FiniteDimensional() category = category.WithBasis().Unital() @@ -224,6 +227,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return V.span_of_basis(b) + @cached_method def _charpoly_coeff(self, i): """ @@ -234,6 +238,19 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): store the trace/determinant (a_{r-1} and a_{0} respectively) separate from the entire characteristic polynomial. """ + if self._basis_normalizers is not None: + # Must be a matrix class? + # WARNING/TODO: this whole mess is mis-designed. + n = self.natural_basis_space().nrows() + field = self.base_ring().base_ring() # yeeeeaaaahhh + J = self.__class__(n, field, False) + (_,x,_,_) = J._charpoly_matrix_system() + p = J._charpoly_coeff(i) + # p might be missing some vars, have to substitute "optionally" + pairs = zip(x.base_ring().gens(), self._basis_normalizers) + substitutions = { v: v*c for (v,c) in pairs } + return p.subs(substitutions) + (A_of_x, x, xr, detA) = self._charpoly_matrix_system() R = A_of_x.base_ring() if i >= self.rank(): @@ -416,9 +433,9 @@ 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.natural_inner_product(X,Y) def is_trivial(self): @@ -537,6 +554,20 @@ 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 +779,30 @@ 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.natural_inner_product(X,Y) + True + + """ + return x.to_vector().inner_product(y.to_vector()) def random_eja(): @@ -802,7 +856,7 @@ def random_eja(): -def _real_symmetric_basis(n, field, normalize): +def _real_symmetric_basis(n, field): """ Return a basis for the space of real symmetric n-by-n matrices. @@ -814,7 +868,7 @@ def _real_symmetric_basis(n, field, normalize): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = _real_symmetric_basis(n, QQbar, False) + sage: B = _real_symmetric_basis(n, QQ) sage: all( M.is_symmetric() for M in B) True @@ -829,13 +883,11 @@ def _real_symmetric_basis(n, field, normalize): Sij = Eij else: Sij = Eij + Eij.transpose() - if normalize: - Sij = Sij / _real_symmetric_matrix_ip(Sij,Sij).sqrt() S.append(Sij) return tuple(S) -def _complex_hermitian_basis(n, field, normalize): +def _complex_hermitian_basis(n, field): """ Returns a basis for the space of complex Hermitian n-by-n matrices. @@ -854,7 +906,7 @@ def _complex_hermitian_basis(n, field, normalize): sage: set_random_seed() sage: n = ZZ.random_element(1,5) sage: field = QuadraticField(2, 'sqrt2') - sage: B = _complex_hermitian_basis(n, field, False) + sage: B = _complex_hermitian_basis(n, field) sage: all( M.is_symmetric() for M in B) True @@ -877,8 +929,7 @@ def _complex_hermitian_basis(n, field, normalize): 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()) @@ -886,15 +937,11 @@ def _complex_hermitian_basis(n, field, normalize): # 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) + return tuple( s.change_ring(field) for s in S ) -def _quaternion_hermitian_basis(n, field, normalize): +def _quaternion_hermitian_basis(n, field): """ Returns a basis for the space of quaternion Hermitian n-by-n matrices. @@ -912,7 +959,7 @@ def _quaternion_hermitian_basis(n, field, normalize): sage: set_random_seed() sage: n = ZZ.random_element(1,5) - sage: B = _quaternion_hermitian_basis(n, QQ, False) + sage: B = _quaternion_hermitian_basis(n, QQ) sage: all( M.is_symmetric() for M in B ) True @@ -1185,7 +1232,10 @@ def _unembed_quaternion_matrix(M): if not n.mod(4).is_zero(): raise ValueError("the matrix 'M' must be a complex embedding") - Q = QuaternionAlgebra(QQ,-1,-1) + # Use the base ring of the matrix to ensure that its entries can be + # multiplied by elements of the quaternion algebra. + field = M.base_ring() + Q = QuaternionAlgebra(field,-1,-1) i,j,k = Q.gens() # Go top-left to bottom-right (reading order), converting every @@ -1199,17 +1249,15 @@ def _unembed_quaternion_matrix(M): raise ValueError('bad on-diagonal submatrix') if submat[0,1] != -submat[1,0].conjugate(): raise ValueError('bad off-diagonal submatrix') - z = submat[0,0].real() + submat[0,0].imag()*i - z += submat[0,1].real()*j + submat[0,1].imag()*k + z = submat[0,0].vector()[0] # real part + z += submat[0,0].vector()[1]*i # imag part + z += submat[0,1].vector()[0]*j # real part + z += submat[0,1].vector()[1]*k # imag part elements.append(z) 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. @@ -1218,15 +1266,6 @@ 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() - -def _complex_hermitian_matrix_ip(X,Y): - # This takes EMBEDDED matrices. - 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 RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ @@ -1291,7 +1330,8 @@ 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:: + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) @@ -1299,8 +1339,11 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: all( b.norm() == 1 for b in J.gens() ) True - Left-multiplication operators are symmetric because they satisfy - the Jordan axiom:: + Since our natural basis is normalized with respect to the natural + inner product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the EJA + the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) @@ -1310,7 +1353,9 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): """ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): - if n > 1: + 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. @@ -1319,8 +1364,11 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): 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_normalizers = tuple( + ~(self.natural_inner_product(s,s).sqrt()) for s in S ) + S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) ) - S = _real_symmetric_basis(n, field, normalize_basis) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(RealSymmetricEJA, self) @@ -1330,10 +1378,6 @@ 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): @@ -1389,7 +1433,8 @@ 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:: + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: sage: set_random_seed() sage: n = ZZ.random_element(1,4) @@ -1397,8 +1442,11 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: all( b.norm() == 1 for b in J.gens() ) True - Left-multiplication operators are symmetric because they satisfy - the Jordan axiom:: + Since our natural basis is normalized with respect to the natural + inner product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the EJA + the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) @@ -1408,7 +1456,9 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): """ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): - if n > 1: + 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. @@ -1417,8 +1467,11 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): 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_normalizers = tuple( + ~(self.natural_inner_product(s,s).sqrt()) for s in S ) + S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) ) - S = _complex_hermitian_basis(n, field, normalize_basis) Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(ComplexHermitianEJA, self) @@ -1429,10 +1482,13 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): **kwargs) - def inner_product(self, x, y): - X = x.natural_representation() - Y = y.natural_representation() - return _complex_hermitian_matrix_ip(X,Y) + @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): @@ -1451,7 +1507,7 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The dimension of this algebra is `n^2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n = ZZ.random_element(1,4) sage: J = QuaternionHermitianEJA(n) sage: J.dimension() == 2*(n^2) - n True @@ -1459,7 +1515,7 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The Jordan multiplication is what we think it is:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n = ZZ.random_element(1,4) sage: J = QuaternionHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() @@ -1480,7 +1536,7 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): Our inner product satisfies the Jordan axiom:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n = ZZ.random_element(1,4) sage: J = QuaternionHermitianEJA(n) sage: x = J.random_element() sage: y = J.random_element() @@ -1488,9 +1544,45 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: (x*y).inner_product(z) == y.inner_product(x*z) True + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,4) + sage: J = QuaternionHermitianEJA(n) + sage: all( b.norm() == 1 for b in J.gens() ) + True + + Since our natural basis is normalized with respect to the natural + inner product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the EJA + the operator is self-adjoint by the Jordan axiom:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: x = QuaternionHermitianEJA(n).random_element() + sage: x.operator().matrix().is_symmetric() + True + """ def __init__(self, n, field=QQ, normalize_basis=True, **kwargs): - S = _quaternion_hermitian_basis(n, field, normalize_basis) + S = _quaternion_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_normalizers = tuple( + ~(self.natural_inner_product(s,s).sqrt()) for s in S ) + S = tuple( s*c for (s,c) in zip(S,self._basis_normalizers) ) + Qs = _multiplication_table_from_matrix_basis(S) fdeja = super(QuaternionHermitianEJA, self) @@ -1500,17 +1592,16 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): natural_basis=S, **kwargs) - def inner_product(self, x, y): - # Since a+bi+cj+dk on the diagonal is represented as - # - # a + bi +cj + dk = [ a b c d] - # [ -b a -d c] - # [ -c d a -b] - # [ -d -c b a], - # - # we'll quadruple-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/4 + @staticmethod + def natural_inner_product(X,Y): + Xu = _unembed_quaternion_matrix(X) + Yu = _unembed_quaternion_matrix(Y) + # The trace need not be real; consider Xu = (i*I) and Yu = I. + # The result will be a quaternion algebra element, which doesn't + # have a "vector" method, but does have coefficient_tuple() method + # that returns the coefficients of 1, i, j, and k -- in that order. + return ((Xu*Yu).trace()).coefficient_tuple()[0] + class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): @@ -1586,4 +1677,27 @@ 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.natural_inner_product(X,Y) + True + + """ + return x.to_vector().inner_product(y.to_vector())