X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=9eba6699819956048747f648fd18bc80847fbbb6;hb=c9db5fa5d2f30833e4da85d32418048d34483e43;hp=a0af3f0ef543dd951b2463a8aef0f295a2561287;hpb=efe6cd067e02b788bfcdc5e8b61e994cd524120c;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index a0af3f0..9eba669 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -64,7 +64,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): mult_table, prefix='e', category=None, - natural_basis=None, + matrix_basis=None, check_field=True, check_axioms=True): """ @@ -115,7 +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() @@ -134,10 +134,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # long run to have the multiplication table be in terms of # algebra elements. We do this after calling the superclass # constructor so that from_vector() knows what to do. - self._multiplication_table = [ - list(map(lambda x: self.from_vector(x), ls)) - for ls in mult_table - ] + self._multiplication_table = [ [ self.vector_space().zero() + for i in range(n) ] + for j in range(n) ] + # take advantage of symmetry + for i in range(n): + for j in range(i+1): + elt = self.from_vector(mult_table[i][j]) + self._multiplication_table[i][j] = elt + self._multiplication_table[j][i] = elt if check_axioms: if not self._is_commutative(): @@ -149,7 +154,7 @@ 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 @@ -182,8 +187,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): 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() @@ -194,7 +199,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x = J.random_element() sage: J(x.to_vector().column()) == x True - """ msg = "not an element of this algebra" if elt == 0: @@ -208,19 +212,17 @@ 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 ) + 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)) @@ -515,18 +517,33 @@ 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, 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. - 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.) + 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. - Note that this will always return a matrix. The standard basis - in `R^n` will be returned as `n`-by-`1` column matrices. + 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:: @@ -538,7 +555,7 @@ 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] @@ -549,36 +566,38 @@ 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 @@ -705,22 +724,22 @@ 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] @@ -975,38 +994,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Ensure that computing the rank actually works, since the ranks of all simple algebras are known and will be cached by default:: - sage: J = HadamardEJA(4) - sage: J.rank.clear_cache() - sage: J.rank() - 4 - - :: - - sage: J = JordanSpinEJA(4) - sage: J.rank.clear_cache() - sage: J.rank() - 2 - - :: - - sage: J = RealSymmetricEJA(3) - sage: J.rank.clear_cache() - sage: J.rank() - 3 - - :: - - sage: J = ComplexHermitianEJA(2) - sage: J.rank.clear_cache() - sage: J.rank() - 2 - - :: + sage: set_random_seed() # long time + sage: J = random_eja() # long time + sage: caches = J.rank() # long time + sage: J.rank.clear_cache() # long time + sage: J.rank() == cached # long time + True - sage: J = QuaternionHermitianEJA(2) - sage: J.rank.clear_cache() - sage: J.rank() - 2 """ return len(self._charpoly_coefficients()) @@ -1031,6 +1025,103 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Element = FiniteDimensionalEuclideanJordanAlgebraElement +class RationalBasisEuclideanJordanAlgebraNg(FiniteDimensionalEuclideanJordanAlgebra): + def __init__(self, + field, + basis, + jordan_product, + inner_product, + orthonormalize=True, + prefix='e', + category=None, + check_field=True, + check_axioms=True): + + n = len(basis) + vector_basis = basis + + from sage.structure.element import is_Matrix + basis_is_matrices = False + + degree = 0 + if n > 0: + if is_Matrix(basis[0]): + basis_is_matrices = True + vector_basis = tuple( map(_mat2vec,basis) ) + degree = basis[0].nrows()**2 + else: + degree = basis[0].degree() + + V = VectorSpace(field, degree) + + # Compute this from "Q" (obtained from Gram-Schmidt) below as + # R = Q.solve_right(A), where the rows of "Q" are the + # orthonormalized vector_basis and and the rows of "A" are the + # original vector_basis. + self._deorthonormalization_matrix = None + + if orthonormalize: + from mjo.eja.eja_utils import gram_schmidt + A = matrix(field, vector_basis) + vector_basis = gram_schmidt(vector_basis, inner_product) + W = V.span_of_basis( vector_basis ) + Q = matrix(field, vector_basis) + # A = QR <==> A.T == R.T*Q.T + # So, Q.solve_right() is equivalent to the Q.T.solve_left() + # that we want. + self._deorthonormalization_matrix = Q.solve_right(A) + + if basis_is_matrices: + from mjo.eja.eja_utils import _vec2mat + basis = tuple( map(_vec2mat,vector_basis) ) + + W = V.span_of_basis( vector_basis ) + + mult_table = [ [0 for i in range(n)] for j in range(n) ] + ip_table = [ [0 for i in range(n)] for j in range(n) ] + + # Note: the Jordan and inner- products are defined in terms + # of the ambient basis. It's important that their arguments + # are in ambient coordinates as well. + for i in range(n): + for j in range(i+1): + # ortho basis w.r.t. ambient coords + q_i = vector_basis[i] + q_j = vector_basis[j] + + if basis_is_matrices: + q_i = _vec2mat(q_i) + q_j = _vec2mat(q_j) + + elt = jordan_product(q_i, q_j) + ip = inner_product(q_i, q_j) + + if basis_is_matrices: + # do another mat2vec because the multiplication + # table is in terms of vectors + elt = _mat2vec(elt) + + elt = W.coordinate_vector(elt) + mult_table[i][j] = elt + mult_table[j][i] = elt + ip_table[i][j] = ip + ip_table[j][i] = ip + + self._inner_product_matrix = matrix(field,ip_table) + + if basis_is_matrices: + for m in basis: + m.set_immutable() + else: + basis = tuple( x.column() for x in basis ) + + super().__init__(field, + mult_table, + prefix, + category, + basis, # matrix basis + check_field, + check_axioms) class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): r""" @@ -1076,7 +1167,7 @@ class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebr return superclass._charpoly_coefficients() mult_table = tuple( - map(lambda x: x.to_vector(), ls) + tuple(map(lambda x: x.to_vector(), ls)) for ls in self._multiplication_table ) @@ -1106,26 +1197,25 @@ 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 @@ -1187,7 +1277,7 @@ 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. @@ -1208,7 +1298,7 @@ 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 @@ -1221,7 +1311,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): super(MatrixEuclideanJordanAlgebra, self).__init__(field, mult_table, - natural_basis=basis, + matrix_basis=basis, **kwargs) if algebra_dim == 0: @@ -1244,7 +1334,7 @@ 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. @@ -1304,7 +1394,7 @@ 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() @@ -1383,9 +1473,9 @@ 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 @@ -1584,9 +1674,9 @@ 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:: @@ -1601,17 +1691,17 @@ 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, @@ -1652,9 +1742,9 @@ 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 @@ -1879,9 +1969,9 @@ 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:: @@ -1896,17 +1986,17 @@ 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, @@ -1947,9 +2037,9 @@ 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 @@ -2047,7 +2137,7 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, return cls(n, field, **kwargs) -class HadamardEJA(RationalBasisEuclideanJordanAlgebra, +class HadamardEJA(RationalBasisEuclideanJordanAlgebraNg, ConcreteEuclideanJordanAlgebra): """ Return the Euclidean Jordan Algebra corresponding to the set @@ -2090,22 +2180,17 @@ class HadamardEJA(RationalBasisEuclideanJordanAlgebra, """ def __init__(self, n, field=AA, **kwargs): V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] - for i in range(n) ] + basis = V.basis() - # Inner products are real numbers and not algebra - # elements, so once we turn the algebra element - # into a vector in inner_product(), we never go - # back. As a result -- contrary to what we do with - # self._multiplication_table -- we store the inner - # product table as a plain old matrix and not as - # an algebra operator. - ip_table = matrix.identity(field,n) - self._inner_product_matrix = ip_table + def jordan_product(x,y): + return V([ xi*yi for (xi,yi) in zip(x,y) ]) + def inner_product(x,y): + return x.inner_product(y) super(HadamardEJA, self).__init__(field, - mult_table, - check_axioms=False, + basis, + jordan_product, + inner_product, **kwargs) self.rank.set_cache(n) @@ -2130,7 +2215,7 @@ class HadamardEJA(RationalBasisEuclideanJordanAlgebra, return cls(n, field, **kwargs) -class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra, +class BilinearFormEJA(RationalBasisEuclideanJordanAlgebraNg, ConcreteEuclideanJordanAlgebra): r""" The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` @@ -2211,39 +2296,28 @@ class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra, True """ def __init__(self, B, field=AA, **kwargs): - n = B.nrows() - if not B.is_positive_definite(): raise ValueError("bilinear form is not positive-definite") + n = B.nrows() V = VectorSpace(field, n) - mult_table = [[V.zero() for j in range(n)] for i in range(n)] - for i in range(n): - for j in range(n): - x = V.gen(i) - y = V.gen(j) - x0 = x[0] - xbar = x[1:] - y0 = y[0] - ybar = y[1:] - z0 = (B*x).inner_product(y) - zbar = y0*xbar + x0*ybar - z = V([z0] + zbar.list()) - mult_table[i][j] = z - - # Inner products are real numbers and not algebra - # elements, so once we turn the algebra element - # into a vector in inner_product(), we never go - # back. As a result -- contrary to what we do with - # self._multiplication_table -- we store the inner - # product table as a plain old matrix and not as - # an algebra operator. - ip_table = B - self._inner_product_matrix = ip_table + + def inner_product(x,y): + return (B*x).inner_product(y) + + def jordan_product(x,y): + x0 = x[0] + xbar = x[1:] + y0 = y[0] + ybar = y[1:] + z0 = inner_product(x,y) + zbar = y0*xbar + x0*ybar + return V([z0] + zbar.list()) super(BilinearFormEJA, self).__init__(field, - mult_table, - check_axioms=False, + V.basis(), + jordan_product, + inner_product, **kwargs) # The rank of this algebra is two, unless we're in a