X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=7c861834723344ea6403f9f5da289af8aa299ae7;hb=784a0dc84c5a735c16d604ead783899f5d718c51;hp=00a15a1c56897172f57aeec2d43a391f3b367a45;hpb=8b70663d4c5e51aa5bd0a567c289f67e5ff8c000;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 00a15a1..7c86183 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -1,6 +1,6 @@ -from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement from sage.matrix.constructor import matrix from sage.modules.free_module import VectorSpace +from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement # TODO: make this unnecessary somehow. from sage.misc.lazy_import import lazy_import @@ -10,7 +10,7 @@ lazy_import('mjo.eja.eja_subalgebra', from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator from mjo.eja.eja_utils import _mat2vec -class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraElement): +class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): """ An element of a Euclidean Jordan algebra. """ @@ -25,68 +25,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle dir(self.__class__) ) - def __init__(self, A, elt=None): - """ - - SETUP:: - - sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, - ....: random_eja) - - EXAMPLES: - - The identity in `S^n` is converted to the identity in the EJA:: - - sage: J = RealSymmetricEJA(3) - sage: I = matrix.identity(QQ,3) - sage: J(I) == J.one() - True - - This skew-symmetric matrix can't be represented in the EJA:: - - sage: J = RealSymmetricEJA(3) - sage: A = matrix(QQ,3, lambda i,j: i-j) - sage: J(A) - Traceback (most recent call last): - ... - ArithmeticError: vector is not in free module - - TESTS: - - Ensure that we can convert any element of the parent's - underlying vector space back into an algebra element whose - vector representation is what we started with:: - sage: set_random_seed() - sage: J = random_eja() - sage: v = J.vector_space().random_element() - sage: J(v).vector() == v - True - - """ - # Goal: if we're given a matrix, and if it lives in our - # parent algebra's "natural ambient space," convert it - # into an algebra element. - # - # The catch is, we make a recursive call after converting - # the given matrix into a vector that lives in the algebra. - # This we need to try the parent class initializer first, - # to avoid recursing forever if we're given something that - # already fits into the algebra, but also happens to live - # in the parent's "natural ambient space" (this happens with - # vectors in R^n). - try: - FiniteDimensionalAlgebraElement.__init__(self, A, elt) - except ValueError: - natural_basis = A.natural_basis() - if elt in natural_basis[0].matrix_space(): - # 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()**2) - W = V.span( _mat2vec(s) for s in natural_basis ) - coords = W.coordinates(_mat2vec(elt)) - FiniteDimensionalAlgebraElement.__init__(self, A, coords) def __pow__(self, n): """ @@ -139,7 +78,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle elif n == 1: return self else: - return (self.operator()**(n-1))(self) + return (self**(n-1))*self def apply_univariate_polynomial(self, p): @@ -226,9 +165,24 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: x.apply_univariate_polynomial(p) 0 + 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: p1 = J.one().characteristic_polynomial() + sage: q1 = J.zero().characteristic_polynomial() + sage: e0,e1,e2 = J.gens() + sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3 + sage: p2 = A.one().characteristic_polynomial() + sage: q2 = A.zero().characteristic_polynomial() + sage: p1 == p2 + True + sage: q1 == q2 + True + """ p = self.parent().characteristic_polynomial() - return p(*self.vector()) + return p(*self.to_vector()) def inner_product(self, other): @@ -255,7 +209,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) 32 - sage: J(x).inner_product(J(y)) + sage: J.from_vector(x).inner_product(J.from_vector(y)) 32 The inner product on `S^n` is ` = trace(X*Y)`, where @@ -289,9 +243,8 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: x.inner_product(y) in RR + sage: x,y = J.random_elements(2) + sage: x.inner_product(y) in RLF True """ @@ -326,9 +279,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Test Lemma 1 from Chapter III of Koecher:: sage: set_random_seed() - sage: J = random_eja() - sage: u = J.random_element() - sage: v = J.random_element() + sage: u,v = random_eja().random_elements(2) sage: lhs = u.operator_commutes_with(u*v) sage: rhs = v.operator_commutes_with(u^2) sage: lhs == rhs @@ -338,9 +289,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Chapter III, or from Baes (2.3):: sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() + sage: x,y = random_eja().random_elements(2) sage: Lx = x.operator() sage: Ly = y.operator() sage: Lxx = (x*x).operator() @@ -352,10 +301,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Baes (2.4):: 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 = random_eja().random_elements(3) sage: Lx = x.operator() sage: Ly = y.operator() sage: Lz = z.operator() @@ -369,10 +315,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Baes (2.5):: sage: set_random_seed() - sage: J = random_eja() - sage: u = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() + sage: u,y,z = random_eja().random_elements(3) sage: Lu = u.operator() sage: Ly = y.operator() sage: Lz = z.operator() @@ -429,6 +372,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: x.is_invertible() == (x.det() != 0) True + Ensure that the determinant is multiplicative on an associative + subalgebra as in Faraut and Koranyi's Proposition II.2.2:: + + sage: set_random_seed() + sage: J = random_eja().random_element().subalgebra_generated_by() + sage: x,y = J.random_elements(2) + sage: (x*y).det() == x.det()*y.det() + True + """ P = self.parent() r = P.rank() @@ -437,7 +389,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle # -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. - return ((-1)**r)*p(*self.vector()) + return ((-1)**r)*p(*self.to_vector()) def inverse(self): @@ -460,18 +412,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Example 11.11:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) + sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: while not x.is_invertible(): ....: x = J.random_element() - sage: x_vec = x.vector() + sage: x_vec = x.to_vector() sage: x0 = x_vec[0] 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: x.inverse() == J(x_inverse) + sage: x.inverse() == J.from_vector(x_inverse) True TESTS: @@ -543,15 +494,23 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: J.one().is_invertible() True - The zero element is never invertible:: + The zero element is never invertible in a non-trivial algebra:: sage: set_random_seed() sage: J = random_eja() - sage: J.zero().is_invertible() + sage: (not J.is_trivial()) and J.zero().is_invertible() False """ - zero = self.parent().zero() + if self.is_zero(): + if self.parent().is_trivial(): + return True + else: + return False + + # In fact, we only need to know if the constant term is non-zero, + # so we can pass in the field's zero element instead. + zero = self.base_ring().zero() p = self.minimal_polynomial() return not (p(zero) == zero) @@ -679,8 +638,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle aren't multiples of the identity are regular:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) + sage: J = JordanSpinEJA.random_instance() sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 True @@ -704,6 +662,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle True """ + if self.is_zero() and not self.parent().is_trivial(): + # The minimal polynomial of zero in a nontrivial algebra + # is "t"; in a trivial algebra it's "1" by convention + # (it's an empty product). + return 1 return self.subalgebra_generated_by().dimension() @@ -732,6 +695,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA, ....: random_eja) TESTS: @@ -757,16 +721,18 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle The minimal polynomial and the characteristic polynomial coincide and are known (see Alizadeh, Example 11.11) for all elements of the spin factor algebra that aren't scalar multiples of the - identity:: + identity. We require the dimension of the algebra to be at least + two here so that said elements actually exist:: sage: set_random_seed() - sage: n = ZZ.random_element(2,10) + sage: n_max = max(2, JordanSpinEJA._max_test_case_size()) + sage: n = ZZ.random_element(2, n_max) sage: J = JordanSpinEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() - sage: y0 = y.vector()[0] - sage: y_bar = y.vector()[1:] + sage: y0 = y.to_vector()[0] + sage: y_bar = y.to_vector()[1:] sage: actual = y.minimal_polynomial() sage: t = PolynomialRing(J.base_ring(),'t').gen(0) sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2) @@ -781,7 +747,34 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: x.apply_univariate_polynomial(p) 0 + The minimal polynomial is invariant under a change of basis, + and in particular, a re-scaling of the basis:: + + 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,False) + sage: X = random_matrix(QQ,n) + sage: X = X*X.transpose() + sage: x1 = J1(X) + sage: x2 = J2(X) + sage: x1.minimal_polynomial() == x2.minimal_polynomial() + True + """ + if self.is_zero(): + # We would generate a zero-dimensional subalgebra + # where the minimal polynomial would be constant. + # That might be correct, but only if *this* algebra + # is trivial too. + if not self.parent().is_trivial(): + # Pretty sure we know what the minimal polynomial of + # the zero operator is going to be. This ensures + # consistency of e.g. the polynomial variable returned + # in the "normal" case without us having to think about it. + return self.operator().minimal_polynomial() + A = self.subalgebra_generated_by() return A(self).operator().minimal_polynomial() @@ -806,7 +799,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: J = ComplexHermitianEJA(3) sage: J.one() - e0 + e5 + e8 + e0 + e3 + e8 sage: J.one().natural_representation() [1 0 0 0 0 0] [0 1 0 0 0 0] @@ -819,7 +812,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: J = QuaternionHermitianEJA(3) sage: J.one() - e0 + e9 + e14 + e0 + e5 + e14 sage: J.one().natural_representation() [1 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0 0] @@ -836,8 +829,35 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle """ B = self.parent().natural_basis() - W = B[0].matrix_space() - return W.linear_combination(zip(self.vector(), B)) + W = self.parent().natural_basis_space() + return W.linear_combination(zip(B,self.to_vector())) + + + def norm(self): + """ + The norm of this element with respect to :meth:`inner_product`. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealCartesianProductEJA) + + EXAMPLES:: + + sage: J = RealCartesianProductEJA(2) + sage: x = sum(J.gens()) + sage: x.norm() + sqrt(2) + + :: + + sage: J = JordanSpinEJA(4) + sage: x = sum(J.gens()) + sage: x.norm() + 2 + + """ + return self.inner_product(self).sqrt() def operator(self): @@ -853,8 +873,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle 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.operator()(y) == x*y True sage: y.operator()(x) == x*y @@ -862,11 +881,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle """ P = self.parent() - fda_elt = FiniteDimensionalAlgebraElement(P, self) + left_mult_by_self = lambda y: self*y + L = P.module_morphism(function=left_mult_by_self, codomain=P) return FiniteDimensionalEuclideanJordanAlgebraOperator( P, P, - fda_elt.matrix().transpose() ) + L.matrix() ) def quadratic_representation(self, other=None): @@ -884,10 +904,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Alizadeh's Example 11.12:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10) - sage: J = JordanSpinEJA(n) - sage: x = J.random_element() - sage: x_vec = x.vector() + sage: x = JordanSpinEJA.random_instance().random_element() + sage: x_vec = x.to_vector() + 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)]) @@ -904,8 +923,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle 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: Lx = x.operator() sage: Lxx = (x*x).operator() sage: Qx = x.quadratic_representation() @@ -922,7 +940,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle Property 2 (multiply on the right for :trac:`28272`): - sage: alpha = QQ.random_element() + sage: alpha = J.base_ring().random_element() sage: (alpha*x).quadratic_representation() == Qx*(alpha^2) True @@ -1000,11 +1018,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: from mjo.eja.eja_algebra import random_eja - TESTS:: + TESTS: + + This subalgebra, being composed of only powers, is associative:: sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.subalgebra_generated_by().is_associative() + sage: x0 = random_eja().random_element() + sage: A = x0.subalgebra_generated_by() + sage: x,y,z = A.random_elements(3) + sage: (x*y)*z == x*(y*z) True Squaring in the subalgebra should work the same as in @@ -1016,6 +1038,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: A(x^2) == A(x)*A(x) True + The subalgebra generated by the zero element is trivial:: + + sage: set_random_seed() + sage: A = random_eja().zero().subalgebra_generated_by() + sage: A + Euclidean Jordan algebra of dimension 0 over... + sage: A.one() + 0 + """ return FiniteDimensionalEuclideanJordanElementSubalgebra(self) @@ -1070,7 +1101,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) A = u_next.operator().matrix() - c = J(A.solve_right(u_next.vector())) + c = J.from_vector(A.solve_right(u_next.to_vector())) # Now c is the idempotent we want, but it still lives in the subalgebra. return c.superalgebra_element() @@ -1105,7 +1136,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: set_random_seed() sage: J = random_eja() - sage: J.random_element().trace() in J.base_ring() + sage: J.random_element().trace() in RLF True """ @@ -1116,7 +1147,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle # -1 to ensure that _charpoly_coeff(r-1) is really what # appears in front of t^{r-1} in the charpoly. However, # we want the negative of THAT for the trace. - return -p(*self.vector()) + return -p(*self.to_vector()) def trace_inner_product(self, other): @@ -1129,22 +1160,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle TESTS: - The trace inner product is commutative:: + The trace inner product is commutative, bilinear, and satisfies + the Jordan axiom: sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element(); y = J.random_element() + sage: x,y,z = J.random_elements(3) + sage: # commutative sage: x.trace_inner_product(y) == y.trace_inner_product(x) True - - The trace inner product is bilinear:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() - sage: a = QQ.random_element(); + sage: # bilinear + sage: a = J.base_ring().random_element(); sage: actual = (a*(x+z)).trace_inner_product(y) sage: expected = ( a*x.trace_inner_product(y) + ....: a*z.trace_inner_product(y) ) @@ -1155,15 +1181,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle ....: a*x.trace_inner_product(z) ) sage: actual == expected True - - The trace inner product satisfies the compatibility - condition in the definition of a Euclidean Jordan algebra:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() + sage: # jordan axiom sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z) True @@ -1172,3 +1190,30 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle raise TypeError("'other' must live in the same algebra") return (self*other).trace() + + + def trace_norm(self): + """ + The norm of this element with respect to :meth:`trace_inner_product`. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealCartesianProductEJA) + + EXAMPLES:: + + sage: J = RealCartesianProductEJA(2) + sage: x = sum(J.gens()) + sage: x.trace_norm() + sqrt(2) + + :: + + sage: J = JordanSpinEJA(4) + sage: x = sum(J.gens()) + sage: x.trace_norm() + 2*sqrt(2) + + """ + return self.trace_inner_product(self).sqrt()