X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=287a217e785ff3e44109d3fc8d8fd85ed1ca4771;hb=17aee61574caf7f62a70d181840c2be69879a3e7;hp=107603ce712ecf79d6de89a2e4fd573f17738798;hpb=9c35b8d70e384cd98b8ec7eb7a84cf84db1d1137;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 107603c..287a217 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -25,69 +25,6 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): dir(self.__class__) ) - def __init__(self, A, elt): - """ - - 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).to_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). - ifme = super(FiniteDimensionalEuclideanJordanAlgebraElement, self) - try: - ifme.__init__(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.coordinate_vector(_mat2vec(elt)) - ifme.__init__(A, coords) def __pow__(self, n): @@ -257,7 +194,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): 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 @@ -473,7 +410,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): 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: @@ -553,7 +490,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): False """ - zero = self.parent().zero() + # 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) @@ -785,7 +724,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): """ A = self.subalgebra_generated_by() - return A.element_class(A,self).operator().minimal_polynomial() + return A(self).operator().minimal_polynomial() @@ -808,7 +747,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): 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] @@ -821,7 +760,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): 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] @@ -864,10 +803,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): """ P = self.parent() + left_mult_by_self = lambda y: self*y + L = P.module_morphism(function=left_mult_by_self, codomain=P) return FiniteDimensionalEuclideanJordanAlgebraOperator( P, P, - self.to_matrix() ) + L.matrix() ) def quadratic_representation(self, other=None): @@ -1001,11 +942,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): 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 = A.random_element() + sage: y = A.random_element() + sage: z = A.random_element() + sage: (x*y)*z == x*(y*z) True Squaring in the subalgebra should work the same as in @@ -1071,7 +1018,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) A = u_next.operator().matrix() - c = J(A.solve_right(u_next.to_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()