X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=4e66b2db251274d3091875a20056692f33849513;hb=d1b6531a00fcaec57e4cd1a24b08d3a217712b26;hp=fc64510dde701f7f820456d9972691fe07f71178;hpb=884b2015690a50a12f99852ea82c399309164f22;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index fc64510..4e66b2d 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -438,8 +438,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ Return the matrix space in which this algebra's natural basis elements live. + + 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. """ - if self._natural_basis is None or len(self._natural_basis) == 0: + if self.is_trivial(): + return MatrixSpace(self.base_ring(), 0) + elif 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() @@ -666,7 +673,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): # there's only one. return cls(field) - n = ZZ.random_element(cls._max_test_case_size()) + 1 + n = ZZ.random_element(cls._max_test_case_size() + 1) return cls(n, field, **kwargs) @cached_method @@ -688,6 +695,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): for k in range(n) ) L_x = matrix(F, n, n, L_x_i_j) + + r = None + if self.rank.is_in_cache(): + r = self.rank() + # There's no need to pad the system with redundant + # columns if we *know* they'll be redundant. + n = r + # Compute an extra power in case the rank is equal to # the dimension (otherwise, we would stop at x^(r-1)). x_powers = [ (L_x**k)*self.one().to_vector() @@ -696,7 +711,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): AE = A.extended_echelon_form() E = AE[:,n:] A_rref = AE[:,:n] - r = A_rref.rank() + if r is None: + r = A_rref.rank() b = x_powers[r] # The theory says that only the first "r" coefficients are @@ -894,7 +910,7 @@ class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra): return x.to_vector().inner_product(y.to_vector()) -def random_eja(field=AA, nontrivial=False): +def random_eja(field=AA): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. @@ -908,21 +924,17 @@ def random_eja(field=AA, nontrivial=False): Euclidean Jordan algebra of dimension... """ - eja_classes = [HadamardEJA, - JordanSpinEJA, - RealSymmetricEJA, - ComplexHermitianEJA, - QuaternionHermitianEJA] - if not nontrivial: - eja_classes.append(TrivialEJA) - classname = choice(eja_classes) + classname = choice([TrivialEJA, + HadamardEJA, + JordanSpinEJA, + RealSymmetricEJA, + ComplexHermitianEJA, + QuaternionHermitianEJA]) return classname.random_instance(field=field) - - class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): @staticmethod def _max_test_case_size(): @@ -984,7 +996,26 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): J = MatrixEuclideanJordanAlgebra(QQ, basis, normalize_basis=False) - return J._charpoly_coefficients() + a = J._charpoly_coefficients() + + # Unfortunately, changing the basis does change the + # coefficients of the characteristic polynomial, but since + # these are really the coefficients of the "characteristic + # polynomial of" function, everything is still nice and + # unevaluated. It's therefore "obvious" how scaling the + # basis affects the coordinate variables X1, X2, et + # cetera. Scaling the first basis vector up by "n" adds a + # factor of 1/n into every "X1" term, for example. So here + # we simply undo the basis_normalizer scaling that we + # performed earlier. + # + # The a[0] access here is safe because trivial algebras + # won't have any basis normalizers and therefore won't + # make it to this "else" branch. + XS = a[0].parent().gens() + subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i] + for i in range(len(XS)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) @staticmethod @@ -1002,6 +1033,9 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): # 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. + if len(basis) == 0: + return [] + field = basis[0].base_ring() dimension = basis[0].nrows() @@ -1159,6 +1193,11 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): sage: x.operator().matrix().is_symmetric() True + We can construct the (trivial) algebra of rank zero:: + + sage: RealSymmetricEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ @classmethod def _denormalized_basis(cls, n, field): @@ -1432,6 +1471,11 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): sage: x.operator().matrix().is_symmetric() True + We can construct the (trivial) algebra of rank zero:: + + sage: ComplexHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ @classmethod @@ -1727,6 +1771,11 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): sage: x.operator().matrix().is_symmetric() True + We can construct the (trivial) algebra of rank zero:: + + sage: QuaternionHermitianEJA(0) + Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ @classmethod def _denormalized_basis(cls, n, field):