X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=689a3db016437d1e6eda5c6372e52a3513896671;hb=3a476bd1ea5aef3ecd375e71d50342f1441fd35d;hp=56be0fa327c83d56bd6d9b2f4283b144ee47806f;hpb=e55bb3635ce4f25e45c59e947c144910e03bd09b;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 56be0fa..689a3db 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -5,7 +5,7 @@ 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 itertools import repeat from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra from sage.categories.magmatic_algebras import MagmaticAlgebras @@ -13,6 +13,7 @@ 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.prandom import choice from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace @@ -20,16 +21,35 @@ from sage.rings.all import (ZZ, QQ, RR, RLF, CLF, PolynomialRing, QuadraticField) from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement +lazy_import('mjo.eja.eja_subalgebra', + 'FiniteDimensionalEuclideanJordanSubalgebra') from mjo.eja.eja_utils import _mat2vec class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): - # This is an ugly hack needed to prevent the category framework - # from implementing a coercion from our base ring (e.g. the - # rationals) into the algebra. First of all -- such a coercion is - # nonsense to begin with. But more importantly, it tries to do so - # in the category of rings, and since our algebras aren't - # associative they generally won't be rings. - _no_generic_basering_coercion = True + + def _coerce_map_from_base_ring(self): + """ + Disable the map from the base ring into the algebra. + + Performing a nonsense conversion like this automatically + is counterpedagogical. The fallback is to try the usual + element constructor, which should also fail. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J(1) + Traceback (most recent call last): + ... + ValueError: not a naturally-represented algebra element + + """ + return None def __init__(self, field, @@ -91,8 +111,10 @@ 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 = [ map(lambda x: self.from_vector(x), ls) - for ls in mult_table ] + self._multiplication_table = [ + list(map(lambda x: self.from_vector(x), ls)) + for ls in mult_table + ] def _element_constructor_(self, elt): @@ -106,7 +128,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: RealSymmetricEJA) EXAMPLES: @@ -134,7 +156,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): vector representations) back and forth faithfully:: sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() + sage: J = HadamardEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True @@ -144,15 +166,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ + msg = "not a naturally-represented algebra element" if elt == 0: # The superclass implementation of random_element() # needs to be able to coerce "0" into the algebra. return self.zero() + elif elt in self.base_ring(): + # Ensure that no base ring -> algebra coercion is performed + # by this method. There's some stupidity in sage that would + # otherwise propagate to this method; for example, sage thinks + # 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: - raise ValueError("not a naturally-represented algebra element") + raise ValueError(msg) # Thanks for nothing! Matrix spaces aren't vector spaces in # Sage, so we have to figure out its natural-basis coordinates @@ -251,7 +280,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ (A_of_x, x, xr, detA) = self._charpoly_matrix_system() R = A_of_x.base_ring() - if i >= self.rank(): + + if i == self.rank(): + return R.one() + if i > self.rank(): # Guaranteed by theory return R.zero() @@ -360,7 +392,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA EXAMPLES: @@ -374,12 +406,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: p(*xvec) t^2 - 2*t + 1 + By definition, the characteristic polynomial is a monic + degree-zero polynomial in a rank-zero algebra. Note that + Cayley-Hamilton is indeed satisfied since the polynomial + ``1`` evaluates to the identity element of the algebra on + any argument:: + + sage: J = TrivialEJA() + sage: J.characteristic_polynomial() + 1 + """ r = self.rank() n = self.dimension() - # The list of coefficient polynomials a_1, a_2, ..., a_n. - a = [ self._charpoly_coeff(i) for i in range(n) ] + # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n. + a = [ self._charpoly_coeff(i) for i in range(r+1) ] # We go to a bit of trouble here to reorder the # indeterminates, so that it's easier to evaluate the @@ -391,18 +433,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): S = PolynomialRing(S, R.variable_names()) t = S(t) - # Note: all entries past the rth should be zero. The - # coefficient of the highest power (x^r) is 1, but it doesn't - # appear in the solution vector which contains coefficients - # for the other powers (to make them sum to x^r). - if (r < n): - a[r] = 1 # corresponds to x^r - else: - # When the rank is equal to the dimension, trying to - # assign a[r] goes out-of-bounds. - a.append(1) # corresponds to x^r - - return sum( a[k]*(t**k) for k in xrange(len(a)) ) + return sum( a[k]*(t**k) for k in range(len(a)) ) def inner_product(self, x, y): @@ -442,7 +473,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: TrivialEJA) EXAMPLES:: @@ -450,6 +482,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: J.is_trivial() False + :: + + sage: J = TrivialEJA() + sage: J.is_trivial() + True + """ return self.dimension() == 0 @@ -481,7 +519,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ M = list(self._multiplication_table) # copy - for i in xrange(len(M)): + for i in range(len(M)): # M had better be "square" M[i] = [self.monomial(i)] + M[i] M = [["*"] + list(self.gens())] + M @@ -568,12 +606,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES:: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 @@ -622,6 +660,119 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.linear_combination(zip(self.gens(), coeffs)) + def peirce_decomposition(self, c): + """ + The Peirce decomposition of this algebra relative to the + idempotent ``c``. + + In the future, this can be extended to a complete system of + orthogonal idempotents. + + INPUT: + + - ``c`` -- an idempotent of this algebra. + + OUTPUT: + + A triple (J0, J5, J1) containing two subalgebras and one subspace + of this algebra, + + - ``J0`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue zero. + + - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()`` + corresponding to the eigenvalue one-half. + + - ``J1`` -- the algebra on the eigenspace of ``c.operator()`` + corresponding to the eigenvalue one. + + These are the only possible eigenspaces for that operator, and this + algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are + orthogonal, and are subalgebras of this algebra with the appropriate + restrictions. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA + + EXAMPLES: + + The canonical example comes from the symmetric matrices, which + decompose into diagonal and off-diagonal parts:: + + sage: J = RealSymmetricEJA(3) + sage: C = matrix(QQ, [ [1,0,0], + ....: [0,1,0], + ....: [0,0,0] ]) + sage: c = J(C) + sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: J0 + Euclidean Jordan algebra of dimension 1... + sage: J5 + Vector space of degree 6 and dimension 2... + sage: J1 + Euclidean Jordan algebra of dimension 3... + + TESTS: + + Every algebra decomposes trivially with respect to its identity + element:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J0,J5,J1 = J.peirce_decomposition(J.one()) + sage: J0.dimension() == 0 and J5.dimension() == 0 + True + sage: J1.superalgebra() == J and J1.dimension() == J.dimension() + True + + The identity elements in the two subalgebras are the + projections onto their respective subspaces of the + superalgebra's identity element:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: if not J.is_trivial(): + ....: while x.is_nilpotent(): + ....: x = J.random_element() + sage: c = x.subalgebra_idempotent() + sage: J0,J5,J1 = J.peirce_decomposition(c) + sage: J1(c) == J1.one() + True + sage: J0(J.one() - c) == J0.one() + True + + """ + if not c.is_idempotent(): + raise ValueError("element is not idempotent: %s" % c) + + # 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 + # eigenspace corresponding to lambda=1 if you take the + # decomposition relative to the identity element). + trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ()) + J0 = trivial # eigenvalue zero + J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half + J1 = trivial # eigenvalue one + + for (eigval, eigspace) in c.operator().matrix().left_eigenspaces(): + if eigval == ~(self.base_ring()(2)): + J5 = eigspace + else: + gens = tuple( self.from_vector(b) for b in eigspace.basis() ) + subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens) + if eigval == 0: + J0 = subalg + elif eigval == 1: + J1 = subalg + else: + raise ValueError("unexpected eigenvalue: %s" % eigval) + + return (J0, J5, J1) + + def random_elements(self, count): """ Return ``count`` random elements as a tuple. @@ -640,7 +791,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ - return tuple( self.random_element() for idx in xrange(count) ) + return tuple( self.random_element() for idx in range(count) ) def rank(self): @@ -767,12 +918,16 @@ class KnownRankEJA(object): Beware, this will crash for "most instances" because the constructor below looks wrong. """ + if cls is TrivialEJA: + # The TrivialEJA class doesn't take an "n" argument because + # there's only one. + return cls(field) + n = ZZ.random_element(cls._max_test_case_size()) + 1 return cls(n, field, **kwargs) -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA): +class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. @@ -783,13 +938,13 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA EXAMPLES: This multiplication table can be verified by hand:: - sage: J = RealCartesianProductEJA(3) + sage: J = HadamardEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 @@ -808,16 +963,16 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, We can change the generator prefix:: - sage: RealCartesianProductEJA(3, prefix='r').gens() + sage: HadamardEJA(3, prefix='r').gens() (r0, r1, r2) """ def __init__(self, n, field=QQ, **kwargs): V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in xrange(n) ] - for i in xrange(n) ] + mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] + for i in range(n) ] - fdeja = super(RealCartesianProductEJA, self) + fdeja = super(HadamardEJA, self) return fdeja.__init__(field, mult_table, rank=n, **kwargs) def inner_product(self, x, y): @@ -826,7 +981,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA TESTS: @@ -834,7 +989,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, over `R^n`:: sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() + sage: J = HadamardEJA.random_instance() sage: x,y = J.random_elements(2) sage: X = x.natural_representation() sage: Y = y.natural_representation() @@ -845,32 +1000,10 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, return x.to_vector().inner_product(y.to_vector()) -def random_eja(field=QQ): +def random_eja(field=QQ, nontrivial=False): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. - ALGORITHM: - - For now, we choose a random natural number ``n`` (greater than zero) - and then give you back one of the following: - - * The cartesian product of the rational numbers ``n`` times; this is - ``QQ^n`` with the Hadamard product. - - * The Jordan spin algebra on ``QQ^n``. - - * The ``n``-by-``n`` rational symmetric matrices with the symmetric - product. - - * The ``n``-by-``n`` complex-rational Hermitian matrices embedded - in the space of ``2n``-by-``2n`` real symmetric matrices. - - * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded - in the space of ``4n``-by-``4n`` real symmetric matrices. - - Later this might be extended to return Cartesian products of the - EJAs above. - SETUP:: sage: from mjo.eja.eja_algebra import random_eja @@ -881,7 +1014,10 @@ def random_eja(field=QQ): Euclidean Jordan algebra of dimension... """ - classname = choice(KnownRankEJA.__subclasses__()) + eja_classes = KnownRankEJA.__subclasses__() + if nontrivial: + eja_classes.remove(TrivialEJA) + classname = choice(eja_classes) return classname.random_instance(field=field) @@ -921,7 +1057,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): 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)) + basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers)) Qs = self.multiplication_table_from_matrix_basis(basis) @@ -944,8 +1080,8 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # 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) ) + basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + self._basis_normalizers) ) # Do this over the rationals and convert back at the end. J = MatrixEuclideanJordanAlgebra(QQ, @@ -955,7 +1091,7 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): (_,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) + pairs = zip(x.base_ring().gens(), self._basis_normalizers) substitutions = { v: v*c for (v,c) in pairs } result = p.subs(substitutions) @@ -988,9 +1124,9 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): 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): + 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)) @@ -1161,8 +1297,8 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA): # 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): + for i in range(n): + for j in range(i+1): Eij = matrix(field, n, lambda k,l: k==i and l==j) if i == j: Sij = Eij @@ -1292,8 +1428,8 @@ class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): # 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): + for k in range(n/2): + for j in range(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') @@ -1444,8 +1580,8 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA): # * The diagonal will (as a result) be real. # S = [] - for i in xrange(n): - for j in xrange(i+1): + for i in range(n): + for j in range(i+1): Eij = matrix(F, n, lambda k,l: k==i and l==j) if i == j: Sij = cls.real_embed(Eij) @@ -1582,8 +1718,8 @@ class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra): # 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): + for l in range(n/4): + for m in range(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(): @@ -1735,8 +1871,8 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, # * The diagonal will (as a result) be real. # S = [] - for i in xrange(n): - for j in xrange(i+1): + for i in range(n): + for j in range(i+1): Eij = matrix(Q, n, lambda k,l: k==i and l==j) if i == j: Sij = cls.real_embed(Eij) @@ -1763,11 +1899,131 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) +class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): + r""" + The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` + with the half-trace inner product and jordan product ``x*y = + (x0*y0 + , x0*y_bar + y0*x_bar)`` where ``B`` is a + symmetric positive-definite "bilinear form" matrix. It has + dimension `n` over the reals, and reduces to the ``JordanSpinEJA`` + when ``B`` is the identity matrix of order ``n-1``. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) + + EXAMPLES: + + When no bilinear form is specified, the identity matrix is used, + and the resulting algebra is the Jordan spin algebra:: + + sage: J0 = BilinearFormEJA(3) + sage: J1 = JordanSpinEJA(3) + sage: J0.multiplication_table() == J0.multiplication_table() + True + + TESTS: + + We can create a zero-dimensional algebra:: + + sage: J = BilinearFormEJA(0) + sage: J.basis() + Finite family {} + + We can check the multiplication condition given in the Jordan, von + Neumann, and Wigner paper (and also discussed on my "On the + symmetry..." paper). Note that this relies heavily on the standard + choice of basis, as does anything utilizing the bilinear form matrix:: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B = M.transpose()*M + sage: J = BilinearFormEJA(n, B=B) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list())) + ....: for ei in eis ] + sage: actual = [ sis[i]*sis[j] + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: expected = [ J.one() if i == j else J.zero() + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: actual == expected + True + """ + def __init__(self, n, field=QQ, B=None, **kwargs): + if B is None: + self._B = matrix.identity(field, max(0,n-1)) + else: + self._B = B + + 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 = x0*y0 + (self._B*xbar).inner_product(ybar) + zbar = y0*xbar + x0*ybar + z = V([z0] + zbar.list()) + mult_table[i][j] = z + + # The rank of this algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded + # by the ambient dimension). + fdeja = super(BilinearFormEJA, self) + return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) + + def inner_product(self, x, y): + r""" + Half of the trace inner product. + + This is defined so that the special case of the Jordan spin + algebra gets the usual inner product. + + SETUP:: + + sage: from mjo.eja.eja_algebra import BilinearFormEJA + + TESTS: + + Ensure that this is one-half of the trace inner-product:: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, n-1, algorithm='unimodular') + sage: B = M.transpose()*M + sage: J = BilinearFormEJA(n, B=B) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list())) + ....: for ei in eis ] + sage: actual = [ sis[i]*sis[j] + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: expected = [ J.one() if i == j else J.zero() + ....: for i in range(n-1) + ....: for j in range(n-1) ] + + """ + xvec = x.to_vector() + xbar = xvec[1:] + yvec = y.to_vector() + ybar = yvec[1:] + return x[0]*y[0] + (self._B*xbar).inner_product(ybar) + 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 = - (, x0*y_bar + y0*x_bar)``. It has dimension `n` over + (, x0*y_bar + y0*x_bar)``. It has dimension `n` over the reals. SETUP:: @@ -1803,9 +2059,9 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ def __init__(self, n, field=QQ, **kwargs): V = VectorSpace(field, 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): + 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] @@ -1847,3 +2103,40 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ return x.to_vector().inner_product(y.to_vector()) + + +class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): + """ + The trivial Euclidean Jordan algebra consisting of only a zero element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import TrivialEJA + + EXAMPLES:: + + sage: J = TrivialEJA() + sage: J.dimension() + 0 + sage: J.zero() + 0 + sage: J.one() + 0 + sage: 7*J.one()*12*J.one() + 0 + sage: J.one().inner_product(J.one()) + 0 + sage: J.one().norm() + 0 + sage: J.one().subalgebra_generated_by() + Euclidean Jordan algebra of dimension 0 over Rational Field + sage: J.rank() + 0 + + """ + def __init__(self, field=QQ, **kwargs): + mult_table = [] + fdeja = super(TrivialEJA, self) + # The rank is zero using my definition, namely the dimension of the + # largest subalgebra generated by any element. + return fdeja.__init__(field, mult_table, rank=0, **kwargs)