X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=1cf93cce6127b476796cdc130bfd98cfb7f21e41;hb=7a4d25a9e4be093d2452ffb1e5a9834abcb33553;hp=166c9d217cf71d43fe4bfc0911e6fe6c3e1ec9d2;hpb=5d646c586de50b571d2983b546a05899bf0c20c2;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 166c9d2..1cf93cc 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -1,7 +1,5 @@ # -*- coding: utf-8 -*- -from itertools import izip - from sage.matrix.constructor import matrix from sage.modules.free_module import VectorSpace from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement @@ -9,7 +7,7 @@ from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement # TODO: make this unnecessary somehow. from sage.misc.lazy_import import lazy_import lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra') -lazy_import('mjo.eja.eja_subalgebra', +lazy_import('mjo.eja.eja_element_subalgebra', 'FiniteDimensionalEuclideanJordanElementSubalgebra') from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator from mjo.eja.eja_utils import _mat2vec @@ -98,7 +96,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES:: @@ -106,7 +104,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: R = PolynomialRing(QQ, 't') sage: t = R.gen(0) sage: p = t^4 - t^3 + 5*t - 2 - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one().apply_univariate_polynomial(p) == 3*J.one() True @@ -115,7 +113,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): We should always get back an element of the algebra:: sage: set_random_seed() - sage: p = PolynomialRing(QQ, 't').random_element() + sage: p = PolynomialRing(AA, 't').random_element() sage: J = random_eja() sage: x = J.random_element() sage: x.apply_univariate_polynomial(p) in J @@ -139,7 +137,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA EXAMPLES: @@ -147,14 +145,14 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): the identity element is `(t-1)` from which it follows that the characteristic polynomial should be `(t-1)^3`:: - sage: J = RealCartesianProductEJA(3) + sage: J = HadamardEJA(3) sage: J.one().characteristic_polynomial() t^3 - 3*t^2 + 3*t - 1 Likewise, the characteristic of the zero element in the rank-three algebra `R^{n}` should be `t^{3}`:: - sage: J = RealCartesianProductEJA(3) + sage: J = HadamardEJA(3) sage: J.zero().characteristic_polynomial() t^3 @@ -164,7 +162,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): to zero on that element:: sage: set_random_seed() - sage: x = RealCartesianProductEJA(3).random_element() + sage: x = HadamardEJA(3).random_element() sage: p = x.characteristic_polynomial() sage: x.apply_univariate_polynomial(p) 0 @@ -172,7 +170,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): The characteristic polynomials of the zero and unit elements should be what we think they are in a subalgebra, too:: - sage: J = RealCartesianProductEJA(3) + sage: J = HadamardEJA(3) sage: p1 = J.one().characteristic_polynomial() sage: q1 = J.zero().characteristic_polynomial() sage: e0,e1,e2 = J.gens() @@ -546,6 +544,115 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): return not (p(zero) == zero) + def is_primitive_idempotent(self): + """ + Return whether or not this element is a primitive (or minimal) + idempotent. + + A primitive idempotent is a non-zero idempotent that is not + the sum of two other non-zero idempotents. Remark 2.7.15 in + Baes shows that this is what he refers to as a "minimal + idempotent." + + An element of a Euclidean Jordan algebra is a minimal idempotent + if it :meth:`is_idempotent` and if its Peirce subalgebra + corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes, + Proposition 2.7.17). + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA, + ....: TrivialEJA, + ....: random_eja) + + WARNING:: + + This method is sloooooow. + + EXAMPLES: + + The spectral decomposition of a non-regular element should always + contain at least one non-minimal idempotent:: + + sage: J = RealSymmetricEJA(3) + sage: x = sum(J.gens()) + sage: x.is_regular() + False + sage: [ c.is_primitive_idempotent() + ....: for (l,c) in x.spectral_decomposition() ] + [False, True] + + On the other hand, the spectral decomposition of a regular + element should always be in terms of minimal idempotents:: + + sage: J = JordanSpinEJA(4) + sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) ) + sage: x.is_regular() + True + sage: [ c.is_primitive_idempotent() + ....: for (l,c) in x.spectral_decomposition() ] + [True, True] + + TESTS: + + The identity element is minimal only in an EJA of rank one:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.rank() == 1 or not J.one().is_primitive_idempotent() + True + + A non-idempotent cannot be a minimal idempotent:: + + sage: set_random_seed() + sage: J = JordanSpinEJA(4) + sage: x = J.random_element() + sage: (not x.is_idempotent()) and x.is_primitive_idempotent() + False + + Proposition 2.7.19 in Baes says that an element is a minimal + idempotent if and only if it's idempotent with trace equal to + unity:: + + sage: set_random_seed() + sage: J = JordanSpinEJA(4) + sage: x = J.random_element() + sage: expected = (x.is_idempotent() and x.trace() == 1) + sage: actual = x.is_primitive_idempotent() + sage: actual == expected + True + + Primitive idempotents must be non-zero:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.zero().is_idempotent() + True + sage: J.zero().is_primitive_idempotent() + False + + As a consequence of the fact that primitive idempotents must + be non-zero, there are no primitive idempotents in a trivial + Euclidean Jordan algebra:: + + sage: J = TrivialEJA() + sage: J.one().is_idempotent() + True + sage: J.one().is_primitive_idempotent() + False + + """ + if not self.is_idempotent(): + return False + + if self.is_zero(): + return False + + (_,_,J1) = self.parent().peirce_decomposition(self) + return (J1.dimension() == 1) + + def is_nilpotent(self): """ Return whether or not some power of this element is zero. @@ -802,9 +909,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: n_max = RealSymmetricEJA._max_test_case_size() sage: n = ZZ.random_element(1, n_max) - sage: J1 = RealSymmetricEJA(n,QQ) - sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False) - sage: X = random_matrix(QQ,n) + sage: J1 = RealSymmetricEJA(n) + sage: J2 = RealSymmetricEJA(n,normalize_basis=False) + sage: X = random_matrix(AA,n) sage: X = X*X.transpose() sage: x1 = J1(X) sage: x2 = J2(X) @@ -879,7 +986,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): """ B = self.parent().natural_basis() W = self.parent().natural_basis_space() - return W.linear_combination(izip(B,self.to_vector())) + return W.linear_combination(zip(B,self.to_vector())) def norm(self): @@ -889,14 +996,14 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA) + ....: HadamardEJA) EXAMPLES:: - sage: J = RealCartesianProductEJA(2) + sage: J = HadamardEJA(2) sage: x = sum(J.gens()) sage: x.norm() - sqrt(2) + 1.414213562373095? :: @@ -958,10 +1065,10 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: n = x_vec.degree() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] - sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) + sage: A = matrix(AA, 1, [x_vec.inner_product(x_vec)]) sage: B = 2*x0*x_bar.row() sage: C = 2*x0*x_bar.column() - sage: D = matrix.identity(QQ, n-1) + sage: D = matrix.identity(AA, n-1) sage: D = (x0^2 - x_bar.inner_product(x_bar))*D sage: D = D + 2*x_bar.tensor_product(x_bar) sage: Q = matrix.block(2,2,[A,B,C,D]) @@ -1085,7 +1192,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): The spectral decomposition of the identity is ``1`` times itself, and the spectral decomposition of zero is ``0`` times the identity:: - sage: J = RealSymmetricEJA(3,AA) + sage: J = RealSymmetricEJA(3) sage: J.one() e0 + e2 + e5 sage: J.one().spectral_decomposition() @@ -1095,7 +1202,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS:: - sage: J = RealSymmetricEJA(4,AA) + sage: J = RealSymmetricEJA(4) sage: x = sum(J.gens()) sage: sd = x.spectral_decomposition() sage: l0 = sd[0][0] @@ -1114,7 +1221,6 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): True """ - P = self.parent() A = self.subalgebra_generated_by(orthonormalize_basis=True) result = [] for (evalue, proj) in A(self).operator().spectral_decomposition(): @@ -1180,10 +1286,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: from mjo.eja.eja_algebra import random_eja - TESTS:: + TESTS: + + Ensure that we can find an idempotent in a non-trivial algebra + where there are non-nilpotent elements:: sage: set_random_seed() - sage: J = random_eja() + sage: J = random_eja(nontrivial=True) sage: x = J.random_element() sage: while x.is_nilpotent(): ....: x = J.random_element() @@ -1192,6 +1301,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): True """ + if self.parent().is_trivial(): + return self + if self.is_nilpotent(): raise ValueError("this only works with non-nilpotent elements!") @@ -1202,7 +1314,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): # will be minimal for some natural number s... s = 0 minimal_dim = J.dimension() - for i in xrange(1, minimal_dim): + for i in range(1, minimal_dim): this_dim = (u**i).operator().matrix().image().dimension() if this_dim < minimal_dim: minimal_dim = this_dim @@ -1237,7 +1349,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: TrivialEJA, ....: random_eja) @@ -1255,7 +1367,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): :: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one().trace() 5 @@ -1333,21 +1445,21 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA) + ....: HadamardEJA) EXAMPLES:: - sage: J = RealCartesianProductEJA(2) + sage: J = HadamardEJA(2) sage: x = sum(J.gens()) sage: x.trace_norm() - sqrt(2) + 1.414213562373095? :: sage: J = JordanSpinEJA(4) sage: x = sum(J.gens()) sage: x.trace_norm() - 2*sqrt(2) + 2.828427124746190? """ return self.trace_inner_product(self).sqrt()