X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=02ed966540132c07af408a1a9fb4e6eebc220c31;hb=b05705ab04692f738a57a6ef387662ba5ea46ceb;hp=580c923c37a7c00ed492272161eec70ced95c2de;hpb=ce4c7e206125cd399239698d26f951a4aac073f2;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 580c923..02ed966 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -5,18 +5,22 @@ are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. """ +from itertools import izip, repeat + from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra from sage.categories.magmatic_algebras import MagmaticAlgebras 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.prandom import choice from sage.misc.table import table -from sage.modules.free_module import VectorSpace +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, 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 from sage.structure.element import is_Matrix from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement @@ -49,8 +53,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() + sage: x,y = J.random_elements(2) sage: x*y == y*x True @@ -118,11 +121,11 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): vector representations) back and forth faithfully:: sage: set_random_seed() - sage: J = RealCartesianProductEJA(5) + sage: J = RealCartesianProductEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True - sage: J = JordanSpinEJA(5) + sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True @@ -134,13 +137,17 @@ 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) @@ -207,12 +214,18 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): determinant). """ z = self._a_regular_element() - V = self.vector_space() - V1 = V.span_of_basis( (z**k).to_vector() for k in range(self.rank()) ) + # Don't use the parent vector space directly here in case this + # happens to be a subalgebra. In that case, we would be e.g. + # two-dimensional but span_of_basis() would expect three + # coordinates. + V = VectorSpace(self.base_ring(), self.vector_space().dimension()) + basis = [ (z**k).to_vector() for k in range(self.rank()) ] + V1 = V.span_of_basis( basis ) b = (V1.basis() + V1.complement().basis()) return V.span_of_basis(b) + @cached_method def _charpoly_coeff(self, i): """ @@ -263,7 +276,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # have multivatiate polynomial entries. names = tuple('X' + str(i) for i in range(1,n+1)) R = PolynomialRing(self.base_ring(), names) - V = self.vector_space().change_ring(R) + + # Using change_ring() on the parent's vector space doesn't work + # here because, in a subalgebra, that vector space has a basis + # and change_ring() tries to bring the basis along with it. And + # that doesn't work unless the new ring is a PID, which it usually + # won't be. + V = FreeModule(R,n) # Now let x = (X1,X2,...,Xn) be the vector whose entries are # indeterminates... @@ -370,7 +389,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # assign a[r] goes out-of-bounds. a.append(1) # corresponds to x^r - return sum( a[k]*(t**k) for k in range(len(a)) ) + return sum( a[k]*(t**k) for k in xrange(len(a)) ) def inner_product(self, x, y): @@ -387,21 +406,42 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): EXAMPLES: - The inner product must satisfy its axiom for this algebra to truly - be a Euclidean Jordan Algebra:: + Our inner product is "associative," which means the following for + a symmetric bilinear form:: sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() + sage: x,y,z = J.random_elements(3) sage: (x*y).inner_product(z) == y.inner_product(x*z) 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): + """ + Return whether or not this algebra is trivial. + + A trivial algebra contains only the zero element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + + EXAMPLES:: + + sage: J = ComplexHermitianEJA(3) + sage: J.is_trivial() + False + sage: A = J.zero().subalgebra_generated_by() + sage: A.is_trivial() + True + + """ + return self.dimension() == 0 def multiplication_table(self): @@ -431,7 +471,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ M = list(self._multiplication_table) # copy - for i in range(len(M)): + for i in xrange(len(M)): # M had better be "square" M[i] = [self.monomial(i)] + M[i] M = [["*"] + list(self.gens())] + M @@ -463,8 +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] ) :: @@ -480,11 +520,37 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ if self._natural_basis is None: - return tuple( b.to_vector().column() for b in self.basis() ) + M = self.natural_basis_space() + return tuple( M(b.to_vector()) for b in self.basis() ) else: return self._natural_basis + def natural_basis_space(self): + """ + Return the matrix space in which this algebra's natural basis + elements live. + """ + if self._natural_basis is None or len(self._natural_basis) == 0: + return MatrixSpace(self.base_ring(), self.dimension(), 1) + else: + 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): """ @@ -546,6 +612,35 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.linear_combination(zip(self.gens(), coeffs)) + def random_element(self): + # Temporary workaround for https://trac.sagemath.org/ticket/28327 + if self.is_trivial(): + return self.zero() + else: + s = super(FiniteDimensionalEuclideanJordanAlgebra, self) + return s.random_element() + + def random_elements(self, count): + """ + Return ``count`` random elements as a tuple. + + SETUP:: + + sage: from mjo.eja.eja_algebra import JordanSpinEJA + + EXAMPLES:: + + sage: J = JordanSpinEJA(3) + sage: x,y,z = J.random_elements(3) + sage: all( [ x in J, y in J, z in J ]) + True + sage: len( J.random_elements(10) ) == 10 + True + + """ + return tuple( self.random_element() for idx in xrange(count) ) + + def rank(self): """ Return the rank of this EJA. @@ -579,18 +674,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): The rank of the `n`-by-`n` Hermitian real, complex, or quaternion matrices is `n`:: - sage: RealSymmetricEJA(2).rank() - 2 - sage: ComplexHermitianEJA(2).rank() - 2 + sage: RealSymmetricEJA(4).rank() + 4 + sage: ComplexHermitianEJA(3).rank() + 3 sage: QuaternionHermitianEJA(2).rank() 2 - sage: RealSymmetricEJA(5).rank() - 5 - sage: ComplexHermitianEJA(5).rank() - 5 - sage: QuaternionHermitianEJA(5).rank() - 5 TESTS: @@ -618,7 +707,7 @@ 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() @@ -627,7 +716,57 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Element = FiniteDimensionalEuclideanJordanAlgebraElement -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): +class KnownRankEJA(object): + """ + A class for algebras that we actually know we can construct. The + main issue is that, for most of our methods to make sense, we need + to know the rank of our algebra. Thus we can't simply generate a + "random" algebra, or even check that a given basis and product + satisfy the axioms; because even if everything looks OK, we wouldn't + know the rank we need to actuallty build the thing. + + Not really a subclass of FDEJA because doing that causes method + resolution errors, e.g. + + TypeError: Error when calling the metaclass bases + Cannot create a consistent method resolution + order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra, + KnownRankEJA + + """ + @staticmethod + def _max_test_case_size(): + """ + Return an integer "size" that is an upper bound on the size of + this algebra when it is used in a random test + case. Unfortunately, the term "size" is quite vague -- when + dealing with `R^n` under either the Hadamard or Jordan spin + product, the "size" refers to the dimension `n`. When dealing + with a matrix algebra (real symmetric or complex/quaternion + Hermitian), it refers to the size of the matrix, which is + far less than the dimension of the underlying vector space. + + We default to five in this class, which is safe in `R^n`. The + matrix algebra subclasses (or any class where the "size" is + interpreted to be far less than the dimension) should override + with a smaller number. + """ + return 5 + + @classmethod + def random_instance(cls, field=QQ, **kwargs): + """ + Return a random instance of this type of algebra. + + Beware, this will crash for "most instances" because the + constructor below looks wrong. + """ + n = ZZ.random_element(cls._max_test_case_size()) + 1 + return cls(n, field, **kwargs) + + +class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, + KnownRankEJA): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. @@ -669,14 +808,35 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra): """ def __init__(self, n, field=QQ, **kwargs): V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] - for i in range(n) ] + mult_table = [ [ V.gen(i)*(i == j) for j in xrange(n) ] + for i in xrange(n) ] fdeja = super(RealCartesianProductEJA, self) 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: J = RealCartesianProductEJA.random_instance() + sage: x,y = J.random_elements(2) + 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(): @@ -715,465 +875,441 @@ def random_eja(): Euclidean Jordan algebra of dimension... """ + classname = choice(KnownRankEJA.__subclasses__()) + return classname.random_instance() - # The max_n component lets us choose different upper bounds on the - # value "n" that gets passed to the constructor. This is needed - # because e.g. R^{10} is reasonable to test, while the Hermitian - # 10-by-10 quaternion matrices are not. - (constructor, max_n) = choice([(RealCartesianProductEJA, 6), - (JordanSpinEJA, 6), - (RealSymmetricEJA, 5), - (ComplexHermitianEJA, 4), - (QuaternionHermitianEJA, 3)]) - n = ZZ.random_element(1, max_n) - return constructor(n, field=QQ) -def _real_symmetric_basis(n, field=QQ): - """ - Return a basis for the space of real symmetric n-by-n matrices. - """ - # The basis of symmetric matrices, as matrices, in their R^(n-by-n) - # coordinates. - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) - if i == j: - Sij = Eij - else: - # Beware, orthogonal but not normalized! - Sij = Eij + Eij.transpose() - S.append(Sij) - return tuple(S) - - -def _complex_hermitian_basis(n, field=QQ): - """ - Returns a basis for the space of complex Hermitian n-by-n matrices. - SETUP:: - sage: from mjo.eja.eja_algebra import _complex_hermitian_basis +class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): + @staticmethod + def _max_test_case_size(): + # Play it safe, since this will be squared and the underlying + # field can have dimension 4 (quaternions) too. + return 2 - TESTS:: + def __init__(self, field, basis, rank, normalize_basis=True, **kwargs): + """ + Compared to the superclass constructor, we take a basis instead of + a multiplication table because the latter can be computed in terms + of the former when the product is known (like it is here). + """ + # Used in this class's fast _charpoly_coeff() override. + self._basis_normalizers = None + + # We're going to loop through this a few times, so now's a good + # time to ensure that it isn't a generator expression. + basis = tuple(basis) + + if rank > 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()) + 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 ) + basis = tuple(s*c for (s,c) in izip(basis,self._basis_normalizers)) + + Qs = self.multiplication_table_from_matrix_basis(basis) + + fdeja = super(MatrixEuclideanJordanAlgebra, self) + return fdeja.__init__(field, + Qs, + rank=rank, + natural_basis=basis, + **kwargs) - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) - True - """ - F = QuadraticField(-1, 'I') - I = F.gen() - - # This is like the symmetric case, but we need to be careful: - # - # * We want conjugate-symmetry, not just symmetry. - # * The diagonal will (as a result) be real. - # - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(field, n, lambda k,l: k==i and l==j) - if i == j: - Sij = _embed_complex_matrix(Eij) - S.append(Sij) - else: - # Beware, orthogonal but not normalized! 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()) - S.append(Sij_imag) - return tuple(S) + @cached_method + def _charpoly_coeff(self, i): + """ + Override the parent method with something that tries to compute + over a faster (non-extension) field. + """ + if self._basis_normalizers is None: + # We didn't normalize, so assume that the basis we started + # with had entries in a nice field. + return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i) + else: + basis = ( (b/n) for (b,n) in izip(self.natural_basis(), + self._basis_normalizers) ) + field = self.base_ring().base_ring() # yeeeaahhhhhhh + J = MatrixEuclideanJordanAlgebra(field, + basis, + self.rank(), + normalize_basis=False) + (_,x,_,_) = J._charpoly_matrix_system() + p = J._charpoly_coeff(i) + # p might be missing some vars, have to substitute "optionally" + pairs = izip(x.base_ring().gens(), self._basis_normalizers) + substitutions = { v: v*c for (v,c) in pairs } + return p.subs(substitutions) + + + @staticmethod + def multiplication_table_from_matrix_basis(basis): + """ + At least three of the five simple Euclidean Jordan algebras have the + symmetric multiplication (A,B) |-> (AB + BA)/2, where the + multiplication on the right is matrix multiplication. Given a basis + for the underlying matrix space, this function returns a + multiplication table (obtained by looping through the basis + elements) for an algebra of those matrices. + """ + # In S^2, for example, we nominally have four coordinates even + # though the space is of dimension three only. The vector space V + # is supposed to hold the entire long vector, and the subspace W + # of V will be spanned by the vectors that arise from symmetric + # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. + field = basis[0].base_ring() + dimension = basis[0].nrows() + + V = VectorSpace(field, dimension**2) + W = V.span_of_basis( _mat2vec(s) for s in basis ) + n = len(basis) + mult_table = [[W.zero() for j in xrange(n)] for i in xrange(n)] + for i in xrange(n): + for j in xrange(n): + mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2 + mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry)) + + return mult_table + + + @staticmethod + def real_embed(M): + """ + Embed the matrix ``M`` into a space of real matrices. + The matrix ``M`` can have entries in any field at the moment: + the real numbers, complex numbers, or quaternions. And although + they are not a field, we can probably support octonions at some + point, too. This function returns a real matrix that "acts like" + the original with respect to matrix multiplication; i.e. -def _quaternion_hermitian_basis(n, field=QQ): - """ - Returns a basis for the space of quaternion Hermitian n-by-n matrices. + real_embed(M*N) = real_embed(M)*real_embed(N) - SETUP:: + """ + raise NotImplementedError - sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis - TESTS:: + @staticmethod + def real_unembed(M): + """ + The inverse of :meth:`real_embed`. + """ + raise NotImplementedError + + + @classmethod + def natural_inner_product(cls,X,Y): + Xu = cls.real_unembed(X) + Yu = cls.real_unembed(Y) + tr = (Xu*Yu).trace() + if tr in RLF: + # It's real already. + return tr + + # Otherwise, try the thing that works for complex numbers; and + # if that doesn't work, the thing that works for quaternions. + try: + return tr.vector()[0] # real part, imag part is index 1 + except AttributeError: + # A quaternions 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 tr.coefficient_tuple()[0] + + +class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): + @staticmethod + def real_embed(M): + """ + The identity function, for embedding real matrices into real + matrices. + """ + return M - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) ) - True + @staticmethod + def real_unembed(M): + """ + The identity function, for unembedding real matrices from real + matrices. + """ + return M + +class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): """ - Q = QuaternionAlgebra(QQ,-1,-1) - I,J,K = Q.gens() - - # This is like the symmetric case, but we need to be careful: - # - # * We want conjugate-symmetry, not just symmetry. - # * The diagonal will (as a result) be real. - # - S = [] - for i in xrange(n): - for j in xrange(i+1): - Eij = matrix(Q, n, lambda k,l: k==i and l==j) - if i == j: - Sij = _embed_quaternion_matrix(Eij) - S.append(Sij) - else: - # Beware, orthogonal but not normalized! The second, - # third, and fourth ones have a minus because they're - # conjugated. - Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose()) - S.append(Sij_real) - Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose()) - S.append(Sij_I) - Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose()) - S.append(Sij_J) - Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose()) - S.append(Sij_K) - return tuple(S) - - - -def _multiplication_table_from_matrix_basis(basis): - """ - At least three of the five simple Euclidean Jordan algebras have the - symmetric multiplication (A,B) |-> (AB + BA)/2, where the - multiplication on the right is matrix multiplication. Given a basis - for the underlying matrix space, this function returns a - multiplication table (obtained by looping through the basis - elements) for an algebra of those matrices. - """ - # In S^2, for example, we nominally have four coordinates even - # though the space is of dimension three only. The vector space V - # is supposed to hold the entire long vector, and the subspace W - # of V will be spanned by the vectors that arise from symmetric - # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3. - field = basis[0].base_ring() - dimension = basis[0].nrows() - - V = VectorSpace(field, dimension**2) - W = V.span_of_basis( _mat2vec(s) for s in basis ) - n = len(basis) - mult_table = [[W.zero() for j in range(n)] for i in range(n)] - for i in range(n): - for j in range(n): - mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2 - mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry)) - - return mult_table - - -def _embed_complex_matrix(M): - """ - Embed the n-by-n complex matrix ``M`` into the space of real - matrices of size 2n-by-2n via the map the sends each entry `z = a + - bi` to the block matrix ``[[a,b],[-b,a]]``. + The rank-n simple EJA consisting of real symmetric n-by-n + matrices, the usual symmetric Jordan product, and the trace inner + product. It has dimension `(n^2 + n)/2` over the reals. SETUP:: - sage: from mjo.eja.eja_algebra import _embed_complex_matrix + sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: - sage: F = QuadraticField(-1,'i') - sage: x1 = F(4 - 2*i) - sage: x2 = F(1 + 2*i) - sage: x3 = F(-i) - sage: x4 = F(6) - sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) - sage: _embed_complex_matrix(M) - [ 4 -2| 1 2] - [ 2 4|-2 1] - [-----+-----] - [ 0 -1| 6 0] - [ 1 0| 0 6] + sage: J = RealSymmetricEJA(2) + sage: e0, e1, e2 = J.gens() + sage: e0*e0 + e0 + sage: e1*e1 + 1/2*e0 + 1/2*e2 + sage: e2*e2 + e2 TESTS: - Embedding is a homomorphism (isomorphism, in fact):: + The dimension of this algebra is `(n^2 + n) / 2`:: sage: set_random_seed() - sage: n = ZZ.random_element(5) - 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) - sage: expected = _embed_complex_matrix(X*Y) - sage: actual == expected + sage: n_max = RealSymmetricEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) + sage: J = RealSymmetricEJA(n) + sage: J.dimension() == (n^2 + n)/2 True - """ - n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - field = M.base_ring() - blocks = [] - for z in M.list(): - a = z.real() - b = z.imag() - blocks.append(matrix(field, 2, [[a,b],[-b,a]])) - - # We can drop the imaginaries here. - return matrix.block(field.base_ring(), n, blocks) - - -def _unembed_complex_matrix(M): - """ - The inverse of _embed_complex_matrix(). + The Jordan multiplication is what we think it is:: - SETUP:: + 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: expected = (X*Y + Y*X)/2 + sage: actual == expected + True + sage: J(expected) == x*y + True - sage: from mjo.eja.eja_algebra import (_embed_complex_matrix, - ....: _unembed_complex_matrix) + We can change the generator prefix:: - EXAMPLES:: + sage: RealSymmetricEJA(3, prefix='q').gens() + (q0, q1, q2, q3, q4, q5) - sage: A = matrix(QQ,[ [ 1, 2, 3, 4], - ....: [-2, 1, -4, 3], - ....: [ 9, 10, 11, 12], - ....: [-10, 9, -12, 11] ]) - sage: _unembed_complex_matrix(A) - [ 2*i + 1 4*i + 3] - [ 10*i + 9 12*i + 11] + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: - TESTS: + sage: set_random_seed() + sage: J = RealSymmetricEJA.random_instance() + sage: all( b.norm() == 1 for b in J.gens() ) + True - Unembedding is the inverse of embedding:: + 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: F = QuadraticField(-1, 'i') - sage: M = random_matrix(F, 3) - sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M + sage: x = RealSymmetricEJA.random_instance().random_element() + sage: x.operator().matrix().is_symmetric() True """ - n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(2).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") - - F = QuadraticField(-1, 'i') - i = F.gen() - - # Go top-left to bottom-right (reading order), converting every - # 2-by-2 block we see to a single complex element. - elements = [] - for k in xrange(n/2): - for j in xrange(n/2): - submat = M[2*k:2*k+2,2*j:2*j+2] - if submat[0,0] != submat[1,1]: - raise ValueError('bad on-diagonal submatrix') - if submat[0,1] != -submat[1,0]: - raise ValueError('bad off-diagonal submatrix') - z = submat[0,0] + submat[0,1]*i - elements.append(z) - - return matrix(F, n/2, elements) - - -def _embed_quaternion_matrix(M): - """ - Embed the n-by-n quaternion matrix ``M`` into the space of real - matrices of size 4n-by-4n by first sending each quaternion entry - `z = a + bi + cj + dk` to the block-complex matrix - ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into - a real matrix. + @classmethod + def _denormalized_basis(cls, n, field): + """ + Return a basis for the space of real symmetric n-by-n matrices. - SETUP:: + SETUP:: - sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix + sage: from mjo.eja.eja_algebra import RealSymmetricEJA - EXAMPLES:: + TESTS:: - sage: Q = QuaternionAlgebra(QQ,-1,-1) - sage: i,j,k = Q.gens() - sage: x = 1 + 2*i + 3*j + 4*k - sage: M = matrix(Q, 1, [[x]]) - sage: _embed_quaternion_matrix(M) - [ 1 2 3 4] - [-2 1 -4 3] - [-3 4 1 -2] - [-4 -3 2 1] + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: B = RealSymmetricEJA._denormalized_basis(n,QQ) + sage: all( M.is_symmetric() for M in B) + True - Embedding is a homomorphism (isomorphism, in fact):: + """ + # The basis of symmetric matrices, as matrices, in their R^(n-by-n) + # coordinates. + S = [] + for i in xrange(n): + for j in xrange(i+1): + Eij = matrix(field, n, lambda k,l: k==i and l==j) + if i == j: + Sij = Eij + else: + Sij = Eij + Eij.transpose() + S.append(Sij) + return S - sage: set_random_seed() - sage: n = ZZ.random_element(5) - sage: Q = QuaternionAlgebra(QQ,-1,-1) - sage: X = random_matrix(Q, n) - sage: Y = random_matrix(Q, n) - sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y) - sage: expected = _embed_quaternion_matrix(X*Y) - sage: actual == expected - True - """ - quaternions = M.base_ring() - n = M.nrows() - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - - F = QuadraticField(-1, 'i') - i = F.gen() - - blocks = [] - for z in M.list(): - t = z.coefficient_tuple() - a = t[0] - b = t[1] - c = t[2] - d = t[3] - cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i], - [-c + d*i, a - b*i]]) - blocks.append(_embed_complex_matrix(cplx_matrix)) - - # We should have real entries by now, so use the realest field - # we've got for the return value. - return matrix.block(quaternions.base_ring(), n, blocks) - - -def _unembed_quaternion_matrix(M): - """ - The inverse of _embed_quaternion_matrix(). + @staticmethod + def _max_test_case_size(): + return 4 # Dimension 10 - SETUP:: - sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix, - ....: _unembed_quaternion_matrix) + def __init__(self, n, field=QQ, **kwargs): + basis = self._denormalized_basis(n, field) + super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs) - EXAMPLES:: - sage: M = matrix(QQ, [[ 1, 2, 3, 4], - ....: [-2, 1, -4, 3], - ....: [-3, 4, 1, -2], - ....: [-4, -3, 2, 1]]) - sage: _unembed_quaternion_matrix(M) - [1 + 2*i + 3*j + 4*k] +class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): + @staticmethod + def real_embed(M): + """ + Embed the n-by-n complex matrix ``M`` into the space of real + matrices of size 2n-by-2n via the map the sends each entry `z = a + + bi` to the block matrix ``[[a,b],[-b,a]]``. - TESTS: + SETUP:: - Unembedding is the inverse of embedding:: + sage: from mjo.eja.eja_algebra import \ + ....: ComplexMatrixEuclideanJordanAlgebra - sage: set_random_seed() - sage: Q = QuaternionAlgebra(QQ, -1, -1) - sage: M = random_matrix(Q, 3) - sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M - True + EXAMPLES:: - """ - n = ZZ(M.nrows()) - if M.ncols() != n: - raise ValueError("the matrix 'M' must be square") - if not n.mod(4).is_zero(): - raise ValueError("the matrix 'M' must be a complex embedding") - - Q = QuaternionAlgebra(QQ,-1,-1) - i,j,k = Q.gens() - - # Go top-left to bottom-right (reading order), converting every - # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 - # quaternion block. - elements = [] - for l in xrange(n/4): - for m in xrange(n/4): - submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4]) - if submat[0,0] != submat[1,1].conjugate(): - 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 - 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. -def _matrix_ip(X,Y): - X_mat = X.natural_representation() - Y_mat = Y.natural_representation() - return (X_mat*Y_mat).trace() - - -class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - The rank-n simple EJA consisting of real symmetric n-by-n - matrices, the usual symmetric Jordan product, and the trace inner - product. It has dimension `(n^2 + n)/2` over the reals. + sage: F = QuadraticField(-1, 'i') + sage: x1 = F(4 - 2*i) + sage: x2 = F(1 + 2*i) + sage: x3 = F(-i) + sage: x4 = F(6) + sage: M = matrix(F,2,[[x1,x2],[x3,x4]]) + sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M) + [ 4 -2| 1 2] + [ 2 4|-2 1] + [-----+-----] + [ 0 -1| 6 0] + [ 1 0| 0 6] - SETUP:: + TESTS: - sage: from mjo.eja.eja_algebra import RealSymmetricEJA + Embedding is a homomorphism (isomorphism, in fact):: - EXAMPLES:: + sage: set_random_seed() + sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size() + sage: n = ZZ.random_element(n_max) + sage: F = QuadraticField(-1, 'i') + sage: X = random_matrix(F, n) + sage: Y = random_matrix(F, n) + sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X) + sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y) + sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y) + sage: Xe*Ye == XYe + True - sage: J = RealSymmetricEJA(2) - sage: e0, e1, e2 = J.gens() - sage: e0*e0 - e0 - sage: e1*e1 - e0 + e2 - sage: e2*e2 - e2 + """ + n = M.nrows() + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + field = M.base_ring() + blocks = [] + for z in M.list(): + a = z.vector()[0] # real part, I guess + b = z.vector()[1] # imag part, I guess + blocks.append(matrix(field, 2, [[a,b],[-b,a]])) + + # We can drop the imaginaries here. + return matrix.block(field.base_ring(), n, blocks) + + + @staticmethod + def real_unembed(M): + """ + The inverse of _embed_complex_matrix(). - TESTS: + SETUP:: - The dimension of this algebra is `(n^2 + n) / 2`:: + sage: from mjo.eja.eja_algebra import \ + ....: ComplexMatrixEuclideanJordanAlgebra - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) - sage: J.dimension() == (n^2 + n)/2 - True + EXAMPLES:: - The Jordan multiplication is what we think it is:: + sage: A = matrix(QQ,[ [ 1, 2, 3, 4], + ....: [-2, 1, -4, 3], + ....: [ 9, 10, 11, 12], + ....: [-10, 9, -12, 11] ]) + sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A) + [ 2*i + 1 4*i + 3] + [ 10*i + 9 12*i + 11] - sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = RealSymmetricEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() - sage: actual = (x*y).natural_representation() - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: expected = (X*Y + Y*X)/2 - sage: actual == expected - True - sage: J(expected) == x*y - True + TESTS: - We can change the generator prefix:: + Unembedding is the inverse of embedding:: - sage: RealSymmetricEJA(3, prefix='q').gens() - (q0, q1, q2, q3, q4, q5) + sage: set_random_seed() + sage: F = QuadraticField(-1, 'i') + sage: M = random_matrix(F, 3) + sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M) + sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + True - """ - def __init__(self, n, field=QQ, **kwargs): - S = _real_symmetric_basis(n, field=field) - Qs = _multiplication_table_from_matrix_basis(S) + """ + n = ZZ(M.nrows()) + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + if not n.mod(2).is_zero(): + raise ValueError("the matrix 'M' must be a complex embedding") + + field = M.base_ring() # This should already have sqrt2 + R = PolynomialRing(field, '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 + # 2-by-2 block we see to a single complex element. + elements = [] + for k in xrange(n/2): + for j in xrange(n/2): + submat = M[2*k:2*k+2,2*j:2*j+2] + if submat[0,0] != submat[1,1]: + raise ValueError('bad on-diagonal submatrix') + if submat[0,1] != -submat[1,0]: + raise ValueError('bad off-diagonal submatrix') + z = submat[0,0] + submat[0,1]*i + elements.append(z) + + return matrix(F, n/2, elements) + + + @classmethod + def natural_inner_product(cls,X,Y): + """ + Compute a natural inner product in this algebra directly from + its real embedding. - fdeja = super(RealSymmetricEJA, self) - return fdeja.__init__(field, - Qs, - rank=n, - natural_basis=S, - **kwargs) + SETUP:: - def inner_product(self, x, y): - return _matrix_ip(x,y) + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + + TESTS: + + This gives the same answer as the slow, default method implemented + in :class:`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: X = ComplexHermitianEJA.real_unembed(Xe) + sage: Y = ComplexHermitianEJA.real_unembed(Ye) + sage: expected = (X*Y).trace().vector()[0] + sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye) + sage: actual == expected + True -class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): + """ + return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2 + + +class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): """ The rank-n simple EJA consisting of complex Hermitian n-by-n matrices over the real numbers, the usual symmetric Jordan product, @@ -1189,7 +1325,8 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The dimension of this algebra is `n^2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n_max = ComplexHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) sage: J = ComplexHermitianEJA(n) sage: J.dimension() == n^2 True @@ -1197,10 +1334,8 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The Jordan multiplication is what we think it is:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = ComplexHermitianEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() + 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() @@ -1215,31 +1350,251 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: ComplexHermitianEJA(2, prefix='z').gens() (z0, z1, z2, z3) + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: + + sage: set_random_seed() + sage: J = ComplexHermitianEJA.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 + 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: x = ComplexHermitianEJA.random_instance().random_element() + sage: x.operator().matrix().is_symmetric() + True + """ - def __init__(self, n, field=QQ, **kwargs): - S = _complex_hermitian_basis(n) - Qs = _multiplication_table_from_matrix_basis(S) - fdeja = super(ComplexHermitianEJA, self) - return fdeja.__init__(field, - Qs, - rank=n, - natural_basis=S, - **kwargs) + @classmethod + def _denormalized_basis(cls, n, field): + """ + Returns a basis for the space of complex Hermitian n-by-n matrices. + Why do we embed these? Basically, because all of numerical linear + algebra assumes that you're working with vectors consisting of `n` + entries from a field and scalars from the same field. There's no way + to tell SageMath that (for example) the vectors contain complex + numbers, while the scalar field is real. - def inner_product(self, x, y): - # Since a+bi on the diagonal is represented as + SETUP:: + + sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: field = QuadraticField(2, 'sqrt2') + sage: B = ComplexHermitianEJA._denormalized_basis(n, field) + sage: all( M.is_symmetric() for M in B) + True + + """ + 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: # - # a + bi = [ a b ] - # [ -b a ], + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. # - # we'll double-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/2 + S = [] + for i in xrange(n): + for j in xrange(i+1): + Eij = matrix(F, n, lambda k,l: k==i and l==j) + if i == j: + Sij = cls.real_embed(Eij) + S.append(Sij) + else: + # The second one has a minus because it's conjugated. + Sij_real = cls.real_embed(Eij + Eij.transpose()) + S.append(Sij_real) + Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose()) + S.append(Sij_imag) + + # Since we embedded these, we can drop back to the "field" that we + # started with instead of the complex extension "F". + return ( s.change_ring(field) for s in S ) -class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): + def __init__(self, n, field=QQ, **kwargs): + basis = self._denormalized_basis(n,field) + super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs) + + +class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): + @staticmethod + def real_embed(M): + """ + Embed the n-by-n quaternion matrix ``M`` into the space of real + matrices of size 4n-by-4n by first sending each quaternion entry `z + = a + bi + cj + dk` to the block-complex matrix ``[[a + bi, + c+di],[-c + di, a-bi]]`, and then embedding those into a real + matrix. + + SETUP:: + + sage: from mjo.eja.eja_algebra import \ + ....: QuaternionMatrixEuclideanJordanAlgebra + + EXAMPLES:: + + sage: Q = QuaternionAlgebra(QQ,-1,-1) + sage: i,j,k = Q.gens() + sage: x = 1 + 2*i + 3*j + 4*k + sage: M = matrix(Q, 1, [[x]]) + sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M) + [ 1 2 3 4] + [-2 1 -4 3] + [-3 4 1 -2] + [-4 -3 2 1] + + Embedding is a homomorphism (isomorphism, in fact):: + + sage: set_random_seed() + sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size() + sage: n = ZZ.random_element(n_max) + sage: Q = QuaternionAlgebra(QQ,-1,-1) + sage: X = random_matrix(Q, n) + sage: Y = random_matrix(Q, n) + sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X) + sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y) + sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y) + sage: Xe*Ye == XYe + True + + """ + quaternions = M.base_ring() + n = M.nrows() + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + + F = QuadraticField(-1, 'i') + i = F.gen() + + blocks = [] + for z in M.list(): + t = z.coefficient_tuple() + a = t[0] + b = t[1] + c = t[2] + d = t[3] + cplxM = matrix(F, 2, [[ a + b*i, c + d*i], + [-c + d*i, a - b*i]]) + realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM) + blocks.append(realM) + + # We should have real entries by now, so use the realest field + # we've got for the return value. + return matrix.block(quaternions.base_ring(), n, blocks) + + + + @staticmethod + def real_unembed(M): + """ + The inverse of _embed_quaternion_matrix(). + + SETUP:: + + sage: from mjo.eja.eja_algebra import \ + ....: QuaternionMatrixEuclideanJordanAlgebra + + EXAMPLES:: + + sage: M = matrix(QQ, [[ 1, 2, 3, 4], + ....: [-2, 1, -4, 3], + ....: [-3, 4, 1, -2], + ....: [-4, -3, 2, 1]]) + sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M) + [1 + 2*i + 3*j + 4*k] + + TESTS: + + Unembedding is the inverse of embedding:: + + sage: set_random_seed() + sage: Q = QuaternionAlgebra(QQ, -1, -1) + sage: M = random_matrix(Q, 3) + sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M) + sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M + True + + """ + n = ZZ(M.nrows()) + if M.ncols() != n: + raise ValueError("the matrix 'M' must be square") + if not n.mod(4).is_zero(): + raise ValueError("the matrix 'M' must be a complex embedding") + + # 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 + # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1 + # quaternion block. + elements = [] + for l in xrange(n/4): + for m in xrange(n/4): + submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed( + M[4*l:4*l+4,4*m:4*m+4] ) + if submat[0,0] != submat[1,1].conjugate(): + raise ValueError('bad on-diagonal submatrix') + if submat[0,1] != -submat[1,0].conjugate(): + raise ValueError('bad off-diagonal submatrix') + 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) + + + @classmethod + def natural_inner_product(cls,X,Y): + """ + Compute a natural inner product in this algebra directly from + its real embedding. + + SETUP:: + + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA + + TESTS: + + This gives the same answer as the slow, default method implemented + in :class:`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: 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 == expected + True + + """ + return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4 + + +class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, + KnownRankEJA): """ The rank-n simple EJA consisting of self-adjoint n-by-n quaternion matrices, the usual symmetric Jordan product, and the @@ -1252,10 +1607,11 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): TESTS: - The dimension of this algebra is `n^2`:: + The dimension of this algebra is `2*n^2 - n`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) + sage: n_max = QuaternionHermitianEJA._max_test_case_size() + sage: n = ZZ.random_element(1, n_max) sage: J = QuaternionHermitianEJA(n) sage: J.dimension() == 2*(n^2) - n True @@ -1263,10 +1619,8 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): The Jordan multiplication is what we think it is:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5) - sage: J = QuaternionHermitianEJA(n) - sage: x = J.random_element() - sage: y = J.random_element() + 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() @@ -1281,32 +1635,85 @@ class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: QuaternionHermitianEJA(2, prefix='a').gens() (a0, a1, a2, a3, a4, a5) + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: + + sage: set_random_seed() + sage: J = QuaternionHermitianEJA.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 + 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: x = QuaternionHermitianEJA.random_instance().random_element() + sage: x.operator().matrix().is_symmetric() + True + """ - def __init__(self, n, field=QQ, **kwargs): - S = _quaternion_hermitian_basis(n) - Qs = _multiplication_table_from_matrix_basis(S) + @classmethod + def _denormalized_basis(cls, n, field): + """ + Returns a basis for the space of quaternion Hermitian n-by-n matrices. - fdeja = super(QuaternionHermitianEJA, self) - return fdeja.__init__(field, - Qs, - rank=n, - natural_basis=S, - **kwargs) + Why do we embed these? Basically, because all of numerical + linear algebra assumes that you're working with vectors consisting + of `n` entries from a field and scalars from the same field. There's + no way to tell SageMath that (for example) the vectors contain + complex numbers, while the scalar field is real. - def inner_product(self, x, y): - # Since a+bi+cj+dk on the diagonal is represented as + SETUP:: + + sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA + + TESTS:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ) + sage: all( M.is_symmetric() for M in B ) + True + + """ + Q = QuaternionAlgebra(QQ,-1,-1) + I,J,K = Q.gens() + + # This is like the symmetric case, but we need to be careful: # - # a + bi +cj + dk = [ a b c d] - # [ -b a -d c] - # [ -c d a -b] - # [ -d -c b a], + # * We want conjugate-symmetry, not just symmetry. + # * The diagonal will (as a result) be real. # - # we'll quadruple-count the "a" entries if we take the trace of - # the embedding. - return _matrix_ip(x,y)/4 + S = [] + for i in xrange(n): + for j in xrange(i+1): + Eij = matrix(Q, n, lambda k,l: k==i and l==j) + if i == j: + Sij = cls.real_embed(Eij) + S.append(Sij) + else: + # The second, third, and fourth ones have a minus + # because they're conjugated. + Sij_real = cls.real_embed(Eij + Eij.transpose()) + S.append(Sij_real) + Sij_I = cls.real_embed(I*Eij - I*Eij.transpose()) + S.append(Sij_I) + Sij_J = cls.real_embed(J*Eij - J*Eij.transpose()) + S.append(Sij_J) + Sij_K = cls.real_embed(K*Eij - K*Eij.transpose()) + S.append(Sij_K) + return S -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): + def __init__(self, n, field=QQ, **kwargs): + basis = self._denormalized_basis(n,field) + super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) + + +class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = @@ -1346,9 +1753,9 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra): """ def __init__(self, n, field=QQ, **kwargs): 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): + mult_table = [[V.zero() for j in xrange(n)] for i in xrange(n)] + for i in xrange(n): + for j in xrange(n): x = V.gen(i) y = V.gen(j) x0 = x[0] @@ -1368,4 +1775,25 @@ 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: J = JordanSpinEJA.random_instance() + sage: x,y = J.random_elements(2) + 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())