X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_operator.py;h=2a0c9c48633cd06551c49515840680b57a5b927d;hb=5d646c586de50b571d2983b546a05899bf0c20c2;hp=b225b295c9e0f4e87befb02e28870eb7a0d1b9dd;hpb=9c86e23e2ef79fb9d7003f68952750dda0ca0e0b;p=sage.d.git diff --git a/mjo/eja/eja_operator.py b/mjo/eja/eja_operator.py index b225b29..2a0c9c4 100644 --- a/mjo/eja/eja_operator.py +++ b/mjo/eja/eja_operator.py @@ -1,4 +1,4 @@ -from sage.all import matrix +from sage.matrix.constructor import matrix from sage.categories.all import FreeModules from sage.categories.map import Map @@ -13,6 +13,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): F = domain_eja.base_ring() if not (F == codomain_eja.base_ring()): raise ValueError("domain and codomain must have the same base ring") + if not (F == mat.base_ring()): + raise ValueError("domain and matrix must have the same base ring") # We need to supply something here to avoid getting the # default Homset of the parent FiniteDimensionalAlgebra class, @@ -46,14 +48,14 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): EXAMPLES:: sage: J = JordanSpinEJA(3) - sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens())) + sage: x = J.linear_combination(zip(J.gens(),range(len(J.gens())))) sage: id = identity_matrix(J.base_ring(), J.dimension()) sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) sage: f(x) == x True """ - return self.codomain()(self.matrix()*x.vector()) + return self.codomain().from_vector(self.matrix()*x.to_vector()) def _add_(self, other): @@ -81,17 +83,18 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [2 0 0] [0 2 0] [0 0 2] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... If you try to add two identical vector space operators but on different EJAs, that should blow up:: sage: J1 = RealSymmetricEJA(2) + sage: id1 = identity_matrix(J1.base_ring(), 3) sage: J2 = JordanSpinEJA(3) - sage: id = identity_matrix(QQ, 3) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J1,id) - sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,J2,id) + sage: id2 = identity_matrix(J2.base_ring(), 3) + sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J1,id1) + sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,J2,id2) sage: f + g Traceback (most recent call last): ... @@ -135,8 +138,10 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: [39 54 69] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 1 over Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over + Rational Field + Codomain: Euclidean Jordan algebra of dimension 1 over + Rational Field """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -175,8 +180,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [1 0 0] [0 1 0] [0 0 1] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -209,26 +214,31 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: x.operator() Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: - [ 2 4 0] + [ 2 2 0] [ 2 9 2] - [ 0 4 16] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + [ 0 2 16] + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... sage: x.operator()*(1/2) Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: - [ 1 2 0] + [ 1 1 0] [ 1 9/2 1] - [ 0 2 8] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + [ 0 1 8] + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ - if other in self.codomain().base_ring(): - return FiniteDimensionalEuclideanJordanAlgebraOperator( - self.domain(), - self.codomain(), - self.matrix()*other) + try: + if other in self.codomain().base_ring(): + return FiniteDimensionalEuclideanJordanAlgebraOperator( + self.domain(), + self.codomain(), + self.matrix()*other) + except NotImplementedError: + # This can happen with certain arguments if the base_ring() + # is weird and doesn't know how to test membership. + pass # This should eventually delegate to _composition_ after performing # some sanity checks for us. @@ -256,8 +266,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [-1 0 0] [ 0 -1 0] [ 0 0 -1] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -289,8 +299,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [3 0 0] [0 3 0] [0 0 3] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ if (n == 1): @@ -330,8 +340,10 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): algebras represented by the matrix: [1 0] [0 1] - Domain: Euclidean Jordan algebra of degree 2 over Rational Field - Codomain: Euclidean Jordan algebra of degree 2 over Rational Field + Domain: Euclidean Jordan algebra of dimension 2 over + Rational Field + Codomain: Euclidean Jordan algebra of dimension 2 over + Rational Field """ msg = ("Linear operator between finite-dimensional Euclidean Jordan " @@ -364,13 +376,111 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [-1 0 0] [ 0 -1 0] [ 0 0 -1] - Domain: Euclidean Jordan algebra of degree 3 over Rational Field - Codomain: Euclidean Jordan algebra of degree 3 over Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ return (self + (-other)) + def inverse(self): + """ + Return the inverse of this operator, if it exists. + + The reason this method is not simply an alias for the built-in + :meth:`__invert__` is that the built-in inversion is a bit magic + since it's intended to be a unary operator. If we alias ``inverse`` + to ``__invert__``, then we wind up having to call e.g. ``A.inverse`` + without parentheses. + + SETUP:: + + sage: from mjo.eja.eja_algebra import RealSymmetricEJA, random_eja + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: x = sum(J.gens()) + sage: x.operator().inverse().matrix() + [3/2 -1 1/2] + [ -1 2 -1] + [1/2 -1 3/2] + sage: x.operator().matrix().inverse() + [3/2 -1 1/2] + [ -1 2 -1] + [1/2 -1 3/2] + + TESTS: + + The identity operator is its own inverse:: + + sage: set_random_seed() + sage: J = random_eja() + sage: idJ = J.one().operator() + sage: idJ.inverse() == idJ + True + + The inverse of the inverse is the operator we started with:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: L = x.operator() + sage: not L.is_invertible() or (L.inverse().inverse() == L) + True + + """ + return ~self + + + def is_invertible(self): + """ + Return whether or not this operator is invertible. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, + ....: TrivialEJA, + ....: random_eja) + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: x = sum(J.gens()) + sage: x.operator().matrix() + [ 1 1/2 0] + [1/2 1 1/2] + [ 0 1/2 1] + sage: x.operator().matrix().is_invertible() + True + sage: x.operator().is_invertible() + True + + The zero operator is invertible in a trivial algebra:: + + sage: J = TrivialEJA() + sage: J.zero().operator().is_invertible() + True + + TESTS: + + The identity operator is always invertible:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.one().operator().is_invertible() + True + + The zero operator is never invertible in a nontrivial algebra:: + + sage: set_random_seed() + sage: J = random_eja() + sage: not J.is_trivial() and J.zero().operator().is_invertible() + False + + """ + return self.matrix().is_invertible() + + def matrix(self): """ Return the matrix representation of this operator with respect @@ -414,3 +524,62 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): """ # The matrix method returns a polynomial in 'x' but want one in 't'. return self.matrix().minimal_polynomial().change_variable_name('t') + + + def spectral_decomposition(self): + """ + Return the spectral decomposition of this operator as a list of + (eigenvalue, orthogonal projector) pairs. + + This is the unique spectral decomposition, up to the order of + the projection operators, with distinct eigenvalues. So, the + projections are generally onto subspaces of dimension greater + than one. + + SETUP:: + + sage: from mjo.eja.eja_algebra import RealSymmetricEJA + + EXAMPLES:: + + sage: J = RealSymmetricEJA(4,AA) + sage: x = sum(J.gens()) + sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) + sage: L0x = A(x).operator() + sage: sd = L0x.spectral_decomposition() + sage: l0 = sd[0][0] + sage: l1 = sd[1][0] + sage: P0 = sd[0][1] + sage: P1 = sd[1][1] + sage: P0*l0 + P1*l1 == L0x + True + sage: P0 + P1 == P0^0 # the identity + True + sage: P0^2 == P0 + True + sage: P1^2 == P1 + True + sage: P0*P1 == A.zero().operator() + True + sage: P1*P0 == A.zero().operator() + True + + """ + if not self.matrix().is_symmetric(): + raise ValueError('algebra basis is not orthonormal') + + D,P = self.matrix().jordan_form(subdivide=False,transformation=True) + eigenvalues = D.diagonal() + us = P.columns() + projectors = [] + for i in range(len(us)): + # they won't be normalized, but they have to be + # for the spectral theorem to work. + us[i] = us[i]/us[i].norm() + mat = us[i].column()*us[i].row() + Pi = FiniteDimensionalEuclideanJordanAlgebraOperator( + self.domain(), + self.codomain(), + mat) + projectors.append(Pi) + return zip(eigenvalues, projectors)