X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=2bf7aa2743d1fefc6952e521d17cb1f3d22fa276;hb=ce40356d28ec29ebc9bd883ecc6a79c4f0d18e87;hp=575c2a4754ae261f284c53de4d423011c9bf5418;hpb=c16c11d5b4cd015b6d7516a175a92f46e03bd2d7;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 575c2a4..2bf7aa2 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -1,7 +1,3 @@ -# -*- 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 +5,7 @@ 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 +94,7 @@ 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 +102,7 @@ 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 +111,7 @@ 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 +135,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA EXAMPLES: @@ -147,14 +143,14 @@ 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 +160,7 @@ 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 +168,7 @@ 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() @@ -185,7 +181,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): True """ - p = self.parent().characteristic_polynomial() + p = self.parent().characteristic_polynomial_of() return p(*self.to_vector()) @@ -348,6 +344,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: TrivialEJA, ....: random_eja) EXAMPLES:: @@ -366,6 +363,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.det() -1 + The determinant of the sole element in the rank-zero trivial + algebra is ``1``, by three paths of reasoning. First, its + characteristic polynomial is a constant ``1``, so the constant + term in that polynomial is ``1``. Second, the characteristic + polynomial evaluated at zero is again ``1``. And finally, the + (empty) product of its eigenvalues is likewise just unity:: + + sage: J = TrivialEJA() + sage: J.zero().det() + 1 + TESTS: An element is invertible if and only if its determinant is @@ -384,15 +392,21 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x,y = J.random_elements(2) sage: (x*y).det() == x.det()*y.det() True - """ P = self.parent() r = P.rank() - p = P._charpoly_coeff(0) - # The _charpoly_coeff function already adds the factor of - # -1 to ensure that _charpoly_coeff(0) is really what - # appears in front of t^{0} in the charpoly. However, - # we want (-1)^r times THAT for the determinant. + + if r == 0: + # Special case, since we don't get the a0=1 + # coefficient when the rank of the algebra + # is zero. + return P.base_ring().one() + + p = P._charpoly_coefficients()[0] + # The _charpoly_coeff function already adds the factor of -1 + # to ensure that _charpoly_coefficients()[0] is really what + # appears in front of t^{0} in the charpoly. However, we want + # (-1)^r times THAT for the determinant. return ((-1)**r)*p(*self.to_vector()) @@ -422,11 +436,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: while not x.is_invertible(): ....: x = J.random_element() sage: x_vec = x.to_vector() - sage: x0 = x_vec[0] + sage: x0 = x_vec[:1] sage: x_bar = x_vec[1:] - sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar)) - sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) - sage: x_inverse = coeff*inv_vec + sage: coeff = x0.inner_product(x0) - x_bar.inner_product(x_bar) + sage: x_inverse = x_vec.parent()(x0.list() + (-x_bar).list()) + sage: if not coeff.is_zero(): x_inverse = x_inverse/coeff sage: x.inverse() == J.from_vector(x_inverse) True @@ -546,10 +560,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): return not (p(zero) == zero) - def is_minimal_idempotent(self): + def is_primitive_idempotent(self): """ - Return whether or not this element is a minimal idempotent. + 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 @@ -560,6 +579,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: from mjo.eja.eja_algebra import (JordanSpinEJA, ....: RealSymmetricEJA, + ....: TrivialEJA, ....: random_eja) WARNING:: @@ -571,22 +591,22 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): The spectral decomposition of a non-regular element should always contain at least one non-minimal idempotent:: - sage: J = RealSymmetricEJA(3, AA) + sage: J = RealSymmetricEJA(3) sage: x = sum(J.gens()) sage: x.is_regular() False - sage: [ c.is_minimal_idempotent() + 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, AA) + 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_minimal_idempotent() + sage: [ c.is_primitive_idempotent() ....: for (l,c) in x.spectral_decomposition() ] [True, True] @@ -596,7 +616,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = random_eja() - sage: J.rank() == 1 or not J.one().is_minimal_idempotent() + sage: J.rank() == 1 or not J.one().is_primitive_idempotent() True A non-idempotent cannot be a minimal idempotent:: @@ -604,7 +624,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = JordanSpinEJA(4) sage: x = J.random_element() - sage: (not x.is_idempotent()) and x.is_minimal_idempotent() + sage: (not x.is_idempotent()) and x.is_primitive_idempotent() False Proposition 2.7.19 in Baes says that an element is a minimal @@ -615,24 +635,37 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: J = JordanSpinEJA(4) sage: x = J.random_element() sage: expected = (x.is_idempotent() and x.trace() == 1) - sage: actual = x.is_minimal_idempotent() + 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 + """ - # TODO: when the Peirce decomposition is implemented for real, - # we can use that instead of finding this eigenspace manually. - # - # Trivial eigenspaces don't appear in the list, so we default to the - # trivial one and override it if there's a nontrivial space in the - # list. if not self.is_idempotent(): return False - J1 = VectorSpace(self.parent().base_ring(), 0) - for (eigval, eigspace) in self.operator().matrix().left_eigenspaces(): - if eigval == 1: - J1 = eigspace + if self.is_zero(): + return False + + (_,_,J1) = self.parent().peirce_decomposition(self) return (J1.dimension() == 1) @@ -761,8 +794,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = JordanSpinEJA.random_instance() + sage: n = J.dimension() sage: x = J.random_element() - sage: x == x.coefficient(0)*J.one() or x.degree() == 2 + sage: x.degree() == min(n,2) or (x == x.coefficient(0)*J.one()) True TESTS: @@ -840,13 +874,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: The minimal polynomial of the identity and zero elements are - always the same:: + always the same, except in trivial algebras where the minimal + polynomial of the unit/zero element is ``1``:: sage: set_random_seed() - sage: J = random_eja(nontrivial=True) - sage: J.one().minimal_polynomial() + sage: J = random_eja() + sage: mu = J.one().minimal_polynomial() + sage: t = mu.parent().gen() + sage: mu + int(J.is_trivial())*(t-2) t - 1 - sage: J.zero().minimal_polynomial() + sage: mu = J.zero().minimal_polynomial() + sage: t = mu.parent().gen() + sage: mu + int(J.is_trivial())*(t-1) t The degree of an element is (by one definition) the degree @@ -864,7 +903,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): two here so that said elements actually exist:: sage: set_random_seed() - sage: n_max = max(2, JordanSpinEJA._max_test_case_size()) + sage: n_max = max(2, JordanSpinEJA._max_random_instance_size()) sage: n = ZZ.random_element(2, n_max) sage: J = JordanSpinEJA(n) sage: y = J.random_element() @@ -890,11 +929,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): and in particular, a re-scaling of the basis:: sage: set_random_seed() - sage: n_max = RealSymmetricEJA._max_test_case_size() + sage: n_max = RealSymmetricEJA._max_random_instance_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) @@ -914,7 +953,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): # in the "normal" case without us having to think about it. return self.operator().minimal_polynomial() - A = self.subalgebra_generated_by() + A = self.subalgebra_generated_by(orthonormalize_basis=False) return A(self).operator().minimal_polynomial() @@ -969,7 +1008,7 @@ 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): @@ -979,14 +1018,14 @@ 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? :: @@ -1045,16 +1084,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: x = JordanSpinEJA.random_instance().random_element() sage: x_vec = x.to_vector() + sage: Q = matrix.identity(x.base_ring(), 0) 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: B = 2*x0*x_bar.row() - sage: C = 2*x0*x_bar.column() - sage: D = matrix.identity(QQ, 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]) + sage: if n > 0: + ....: x0 = x_vec[0] + ....: x_bar = x_vec[1:] + ....: A = matrix(x.base_ring(), 1, [x_vec.inner_product(x_vec)]) + ....: B = 2*x0*x_bar.row() + ....: C = 2*x0*x_bar.column() + ....: D = matrix.identity(x.base_ring(), n-1) + ....: D = (x0^2 - x_bar.inner_product(x_bar))*D + ....: D = D + 2*x_bar.tensor_product(x_bar) + ....: Q = matrix.block(2,2,[A,B,C,D]) sage: Q == x.quadratic_representation().matrix() True @@ -1175,7 +1216,7 @@ 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() @@ -1185,7 +1226,7 @@ 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] @@ -1203,8 +1244,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: l0*c0 + l1*c1 == x True + The spectral decomposition should work in subalgebras, too:: + + sage: J = RealSymmetricEJA(4) + sage: (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9) = J.gens() + sage: A = 2*e5 - 2*e8 + sage: (lambda1, c1) = A.spectral_decomposition()[1] + sage: (J0, J5, J1) = J.peirce_decomposition(c1) + sage: (f0, f1, f2) = J1.gens() + sage: f0.spectral_decomposition() + [(0, 1.000000000000000?*f2), (1, 1.000000000000000?*f0)] + """ - P = self.parent() A = self.subalgebra_generated_by(orthonormalize_basis=True) result = [] for (evalue, proj) in A(self).operator().spectral_decomposition(): @@ -1270,18 +1321,25 @@ 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, or that we get the dumb + solution in the trivial algebra:: sage: set_random_seed() sage: J = random_eja() sage: x = J.random_element() - sage: while x.is_nilpotent(): + sage: while x.is_nilpotent() and not J.is_trivial(): ....: x = J.random_element() sage: c = x.subalgebra_idempotent() sage: c^2 == c True """ + if self.parent().is_trivial(): + return self + if self.is_nilpotent(): raise ValueError("this only works with non-nilpotent elements!") @@ -1292,7 +1350,7 @@ 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 @@ -1327,7 +1385,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: TrivialEJA, ....: random_eja) @@ -1345,7 +1403,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): :: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one().trace() 5 @@ -1367,7 +1425,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): # the trace is an empty sum. return P.base_ring().zero() - p = P._charpoly_coeff(r-1) + p = P._charpoly_coefficients()[r-1] # The _charpoly_coeff function already adds the factor of # -1 to ensure that _charpoly_coeff(r-1) is really what # appears in front of t^{r-1} in the charpoly. However, @@ -1423,21 +1481,21 @@ 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()