X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=9d53bc5fdfd8fca177e43e49a187387fbe5a6085;hb=f1ddf1e9eee634161aad87b9c2de0194efb17879;hp=c569098b3d3e5d6fbceef6352e22861dd125a7f8;hpb=af79c1d027cf737d125b11fd41bb0bc2150778fb;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index c569098..9d53bc5 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -24,15 +24,12 @@ from sage.combinat.free_module import CombinatorialFreeModule from sage.matrix.constructor import matrix from sage.matrix.matrix_space import MatrixSpace from sage.misc.cachefunc import cached_method -from sage.misc.lazy_import import lazy_import from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF, PolynomialRing, QuadraticField) from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement -lazy_import('mjo.eja.eja_subalgebra', - 'FiniteDimensionalEuclideanJordanSubalgebra') from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator from mjo.eja.eja_utils import _mat2vec @@ -57,7 +54,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J(1) Traceback (most recent call last): ... - ValueError: not a naturally-represented algebra element + ValueError: not an element of this algebra """ return None @@ -67,7 +64,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): mult_table, prefix='e', category=None, - natural_basis=None, + matrix_basis=None, check_field=True, check_axioms=True): """ @@ -118,7 +115,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): if not all( len(l) == n for l in mult_table ): raise ValueError("multiplication table is not square") - self._natural_basis = natural_basis + self._matrix_basis = matrix_basis if category is None: category = MagmaticAlgebras(field).FiniteDimensional() @@ -152,7 +149,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): def _element_constructor_(self, elt): """ - Construct an element of this algebra from its natural + Construct an element of this algebra from its vector or matrix representation. This gets called only after the parent element _call_ method @@ -180,13 +177,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J(A) Traceback (most recent call last): ... - ArithmeticError: vector is not in free module + ValueError: not an element of this algebra TESTS: Ensure that we can convert any element of the two non-matrix - simple algebras (whose natural representations are their usual - vector representations) back and forth faithfully:: + simple algebras (whose matrix representations are columns) + back and forth faithfully:: sage: set_random_seed() sage: J = HadamardEJA.random_instance() @@ -197,9 +194,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x = J.random_element() sage: J(x.to_vector().column()) == x True - """ - msg = "not a naturally-represented algebra element" + msg = "not an element of this algebra" if elt == 0: # The superclass implementation of random_element() # needs to be able to coerce "0" into the algebra. @@ -211,20 +207,23 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # that the integer 3 belongs to the space of 2-by-2 matrices. raise ValueError(msg) - natural_basis = self.natural_basis() - basis_space = natural_basis[0].matrix_space() - if elt not in basis_space: + if elt not in self.matrix_space(): raise ValueError(msg) # Thanks for nothing! Matrix spaces aren't vector spaces in - # Sage, so we have to figure out its natural-basis coordinates + # Sage, so we have to figure out its matrix-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)) + V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols()) + W = V.span_of_basis( _mat2vec(s) for s in self.matrix_basis() ) + + try: + coords = W.coordinate_vector(_mat2vec(elt)) + except ArithmeticError: # vector is not in free module + raise ValueError(msg) + return self.from_vector(coords) def _repr_(self): @@ -374,6 +373,28 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return (t**r + sum( a[k]*(t**k) for k in range(r) )) + def coordinate_polynomial_ring(self): + r""" + The multivariate polynomial ring in which this algebra's + :meth:`characteristic_polynomial_of` lives. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J = HadamardEJA(2) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2... + sage: J = RealSymmetricEJA(3,QQ) + sage: J.coordinate_polynomial_ring() + Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6... + + """ + var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) ) + return PolynomialRing(self.base_ring(), var_names) def inner_product(self, x, y): """ @@ -491,18 +512,33 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return table(M, header_row=True, header_column=True, frame=True) - def natural_basis(self): + def matrix_basis(self): """ - Return a more-natural representation of this algebra's basis. + Return an (often more natural) representation of this algebras + basis as an ordered tuple of matrices. - Every finite-dimensional Euclidean Jordan Algebra is a direct - sum of five simple algebras, four of which comprise Hermitian - matrices. This method returns the original "natural" basis - for our underlying vector space. (Typically, the natural basis - is used to construct the multiplication table in the first place.) + Every finite-dimensional Euclidean Jordan Algebra is a, up to + Jordan isomorphism, a direct sum of five simple + algebras---four of which comprise Hermitian matrices. And the + last type of algebra can of course be thought of as `n`-by-`1` + column matrices (ambiguusly called column vectors) to avoid + special cases. As a result, matrices (and column vectors) are + a natural representation format for Euclidean Jordan algebra + elements. - Note that this will always return a matrix. The standard basis - in `R^n` will be returned as `n`-by-`1` column matrices. + But, when we construct an algebra from a basis of matrices, + those matrix representations are lost in favor of coordinate + vectors *with respect to* that basis. We could eventually + convert back if we tried hard enough, but having the original + representations handy is valuable enough that we simply store + them and return them from this method. + + Why implement this for non-matrix algebras? Avoiding special + cases for the :class:`BilinearFormEJA` pays with simplicity in + its own right. But mainly, we would like to be able to assume + that elements of a :class:`DirectSumEJA` can be displayed + nicely, without having to have special classes for direct sums + one of whose components was a matrix algebra. SETUP:: @@ -514,7 +550,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = RealSymmetricEJA(2) sage: J.basis() Finite family {0: e0, 1: e1, 2: e2} - sage: J.natural_basis() + sage: J.matrix_basis() ( [1 0] [ 0 0.7071067811865475?] [0 0] [0 0], [0.7071067811865475? 0], [0 1] @@ -525,36 +561,38 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J = JordanSpinEJA(2) sage: J.basis() Finite family {0: e0, 1: e1} - sage: J.natural_basis() + sage: J.matrix_basis() ( [1] [0] [0], [1] ) - """ - if self._natural_basis is None: - M = self.natural_basis_space() + if self._matrix_basis is None: + M = self.matrix_space() return tuple( M(b.to_vector()) for b in self.basis() ) else: - return self._natural_basis + return self._matrix_basis - def natural_basis_space(self): + def matrix_space(self): """ - Return the matrix space in which this algebra's natural basis - elements live. + Return the matrix space in which this algebra's elements live, if + we think of them as matrices (including column vectors of the + appropriate size). Generally this will be an `n`-by-`1` column-vector space, except when the algebra is trivial. There it's `n`-by-`n` - (where `n` is zero), to ensure that two elements of the - natural basis space (empty matrices) can be multiplied. + (where `n` is zero), to ensure that two elements of the matrix + space (empty matrices) can be multiplied. + + Matrix algebras override this with something more useful. """ if self.is_trivial(): return MatrixSpace(self.base_ring(), 0) - elif self._natural_basis is None or len(self._natural_basis) == 0: + elif self._matrix_basis is None or len(self._matrix_basis) == 0: return MatrixSpace(self.base_ring(), self.dimension(), 1) else: - return self._natural_basis[0].matrix_space() + return self._matrix_basis[0].matrix_space() @cached_method @@ -681,22 +719,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Vector space of degree 6 and dimension 2... sage: J1 Euclidean Jordan algebra of dimension 3... - sage: J0.one().natural_representation() + sage: J0.one().to_matrix() [0 0 0] [0 0 0] [0 0 1] sage: orig_df = AA.options.display_format sage: AA.options.display_format = 'radical' - sage: J.from_vector(J5.basis()[0]).natural_representation() + sage: J.from_vector(J5.basis()[0]).to_matrix() [ 0 0 1/2*sqrt(2)] [ 0 0 0] [1/2*sqrt(2) 0 0] - sage: J.from_vector(J5.basis()[1]).natural_representation() + sage: J.from_vector(J5.basis()[1]).to_matrix() [ 0 0 0] [ 0 0 1/2*sqrt(2)] [ 0 1/2*sqrt(2) 0] sage: AA.options.display_format = orig_df - sage: J1.one().natural_representation() + sage: J1.one().to_matrix() [1 0 0] [0 1 0] [0 0 0] @@ -746,6 +784,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): if not c.is_idempotent(): raise ValueError("element is not idempotent: %s" % c) + from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra + # Default these to what they should be if they turn out to be # trivial, because eigenspaces_left() won't return eigenvalues # corresponding to trivial spaces (e.g. it returns only the @@ -854,8 +894,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): of" function. """ n = self.dimension() - var_names = [ "X" + str(z) for z in range(1,n+1) ] - R = PolynomialRing(self.base_ring(), var_names) + R = self.coordinate_polynomial_ring() vars = R.gens() F = R.fraction_field() @@ -1081,26 +1120,25 @@ class ConcreteEuclideanJordanAlgebra: TESTS: - Our natural basis is normalized with respect to the natural inner - product unless we specify otherwise:: + Our basis is normalized with respect to the algebra's inner + product, unless we specify otherwise:: sage: set_random_seed() sage: J = ConcreteEuclideanJordanAlgebra.random_instance() 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 + Since our basis is orthonormal with respect to the algebra's 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:: + 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: J = ConcreteEuclideanJordanAlgebra.random_instance() sage: x = J.random_element() - sage: x.operator().matrix().is_symmetric() + sage: x.operator().is_self_adjoint() True - """ @staticmethod @@ -1162,7 +1200,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt()) basis = tuple( s.change_ring(field) for s in basis ) self._basis_normalizers = tuple( - ~(self.natural_inner_product(s,s).sqrt()) for s in basis ) + ~(self.matrix_inner_product(s,s).sqrt()) for s in basis ) basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers)) # Now compute the multiplication and inner product tables. @@ -1183,7 +1221,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # HACK: ignore the error here if we don't need the # inner product (as is the case when we construct # a dummy QQ-algebra for fast charpoly coefficients. - ip_table[i][j] = self.natural_inner_product(basis[i], + ip_table[i][j] = self.matrix_inner_product(basis[i], basis[j]) except: pass @@ -1196,7 +1234,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): super(MatrixEuclideanJordanAlgebra, self).__init__(field, mult_table, - natural_basis=basis, + matrix_basis=basis, **kwargs) if algebra_dim == 0: @@ -1219,7 +1257,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # entries in a nice field already. Just compute the thing. return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients() - basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + basis = ( (b/n) for (b,n) in zip(self.matrix_basis(), self._basis_normalizers) ) # Do this over the rationals and convert back at the end. @@ -1279,7 +1317,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): raise NotImplementedError @classmethod - def natural_inner_product(cls,X,Y): + def matrix_inner_product(cls,X,Y): Xu = cls.real_unembed(X) Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() @@ -1358,9 +1396,9 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = RealSymmetricEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1559,9 +1597,9 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @classmethod - def natural_inner_product(cls,X,Y): + def matrix_inner_product(cls,X,Y): """ - Compute a natural inner product in this algebra directly from + Compute a matrix inner product in this algebra directly from its real embedding. SETUP:: @@ -1576,17 +1614,17 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: J = ComplexHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: Xe = x.natural_representation() - sage: Ye = y.natural_representation() + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() sage: X = ComplexHermitianEJA.real_unembed(Xe) sage: Y = ComplexHermitianEJA.real_unembed(Ye) sage: expected = (X*Y).trace().real() - sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual = ComplexHermitianEJA.matrix_inner_product(Xe,Ye) sage: actual == expected True """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 + return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/2 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, @@ -1627,9 +1665,9 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = ComplexHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True @@ -1854,9 +1892,9 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): @classmethod - def natural_inner_product(cls,X,Y): + def matrix_inner_product(cls,X,Y): """ - Compute a natural inner product in this algebra directly from + Compute a matrix inner product in this algebra directly from its real embedding. SETUP:: @@ -1871,17 +1909,17 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): sage: set_random_seed() sage: J = QuaternionHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: Xe = x.natural_representation() - sage: Ye = y.natural_representation() + sage: Xe = x.to_matrix() + sage: Ye = y.to_matrix() sage: X = QuaternionHermitianEJA.real_unembed(Xe) sage: Y = QuaternionHermitianEJA.real_unembed(Ye) sage: expected = (X*Y).trace().coefficient_tuple()[0] - sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual = QuaternionHermitianEJA.matrix_inner_product(Xe,Ye) sage: actual == expected True """ - return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 + return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/4 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, @@ -1922,9 +1960,9 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, sage: set_random_seed() sage: J = QuaternionHermitianEJA.random_instance() sage: x,y = J.random_elements(2) - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() + sage: actual = (x*y).to_matrix() + sage: X = x.to_matrix() + sage: Y = y.to_matrix() sage: expected = (X*Y + Y*X)/2 sage: actual == expected True