X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_operator.py;h=0b52f555d51f58341fe56f5b63984f88cdf99da0;hb=4c6ab92d378613a9a053b62382b7e0cda7c450ab;hp=7c3b2a6a4721848caaaf4d30cecbb08d0eab587f;hpb=208bc5d64bd206684e59f757d932768552e9f7ba;p=sage.d.git diff --git a/mjo/eja/eja_operator.py b/mjo/eja/eja_operator.py index 7c3b2a6..0b52f55 100644 --- a/mjo/eja/eja_operator.py +++ b/mjo/eja/eja_operator.py @@ -3,16 +3,54 @@ from sage.categories.all import FreeModules from sage.categories.map import Map class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): + r""" + An operator between two finite-dimensional Euclidean Jordan algebras. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + + EXAMPLES: + + The domain and codomain must be finite-dimensional Euclidean + Jordan algebras; if either is not, then an error is raised:: + + sage: J = HadamardEJA(3) + sage: V = VectorSpace(J.base_ring(), 3) + sage: M = matrix.identity(J.base_ring(), 3) + sage: FiniteDimensionalEuclideanJordanAlgebraOperator(V,J,M) + Traceback (most recent call last): + ... + TypeError: domain must be a finite-dimensional Euclidean + Jordan algebra + sage: FiniteDimensionalEuclideanJordanAlgebraOperator(J,V,M) + Traceback (most recent call last): + ... + TypeError: codomain must be a finite-dimensional Euclidean + Jordan algebra + + """ + def __init__(self, domain_eja, codomain_eja, mat): - # if not ( - # isinstance(domain_eja, FiniteDimensionalEuclideanJordanAlgebra) and - # isinstance(codomain_eja, FiniteDimensionalEuclideanJordanAlgebra) ): - # raise ValueError('(co)domains must be finite-dimensional Euclidean ' - # 'Jordan algebras') + from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra + + # I guess we should check this, because otherwise you could + # pass in pretty much anything algebraish. + if not isinstance(domain_eja, + FiniteDimensionalEuclideanJordanAlgebra): + raise TypeError('domain must be a finite-dimensional ' + 'Euclidean Jordan algebra') + if not isinstance(codomain_eja, + FiniteDimensionalEuclideanJordanAlgebra): + raise TypeError('codomain must be a finite-dimensional ' + 'Euclidean Jordan algebra') 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, @@ -81,19 +119,18 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [2 0 0] [0 2 0] [0 0 2] - Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 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): ... @@ -116,17 +153,17 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator sage: from mjo.eja.eja_algebra import ( ....: JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: RealSymmetricEJA) EXAMPLES:: sage: J1 = JordanSpinEJA(3) - sage: J2 = RealCartesianProductEJA(2) + sage: J2 = HadamardEJA(2) sage: J3 = RealSymmetricEJA(1) - sage: mat1 = matrix(QQ, [[1,2,3], + sage: mat1 = matrix(AA, [[1,2,3], ....: [4,5,6]]) - sage: mat2 = matrix(QQ, [[7,8]]) + sage: mat2 = matrix(AA, [[7,8]]) sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J1, ....: J2, ....: mat1) @@ -138,9 +175,9 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): algebras represented by the matrix: [39 54 69] Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field + Algebraic Real Field Codomain: Euclidean Jordan algebra of dimension 1 over - Rational Field + Algebraic Real Field """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -179,10 +216,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [1 0 0] [0 1 0] [0 0 1] - Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 3 over - Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -215,30 +250,31 @@ 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 dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 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 dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 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. @@ -266,10 +302,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [-1 0 0] [ 0 -1 0] [ 0 0 -1] - Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 3 over - Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ return FiniteDimensionalEuclideanJordanAlgebraOperator( @@ -301,10 +335,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [3 0 0] [0 3 0] [0 0 3] - Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 3 over - Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ if (n == 1): @@ -345,9 +377,9 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [1 0] [0 1] Domain: Euclidean Jordan algebra of dimension 2 over - Rational Field + Algebraic Real Field Codomain: Euclidean Jordan algebra of dimension 2 over - Rational Field + Algebraic Real Field """ msg = ("Linear operator between finite-dimensional Euclidean Jordan " @@ -380,15 +412,178 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): [-1 0 0] [ 0 -1 0] [ 0 0 -1] - Domain: Euclidean Jordan algebra of dimension 3 over - Rational Field - Codomain: Euclidean Jordan algebra of dimension 3 over - Rational Field + Domain: Euclidean Jordan algebra of dimension 3 over... + Codomain: Euclidean Jordan algebra of dimension 3 over... """ return (self + (-other)) + def is_self_adjoint(self): + r""" + Return whether or not this operator is self-adjoint. + + At least in Sage, the fact that the base field is real means + that the algebra elements have to be real as well (this is why + we embed the complex numbers and quaternions). As a result, the + matrix of this operator will contain only real entries, and it + suffices to check only symmetry, not conjugate symmetry. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA) + + EXAMPLES:: + + sage: J = JordanSpinEJA(4) + sage: J.one().operator().is_self_adjoint() + True + + """ + return self.matrix().is_symmetric() + + + def is_zero(self): + r""" + Return whether or not this map is the zero operator. + + SETUP:: + + sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_algebra import (random_eja, + ....: JordanSpinEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J1 = JordanSpinEJA(2) + sage: J2 = RealSymmetricEJA(2) + sage: R = J1.base_ring() + sage: M = matrix(R, [ [0, 0], + ....: [0, 0], + ....: [0, 0] ]) + sage: L = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J2,M) + sage: L.is_zero() + True + sage: M = matrix(R, [ [0, 0], + ....: [0, 1], + ....: [0, 0] ]) + sage: L = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J2,M) + sage: L.is_zero() + False + + TESTS: + + The left-multiplication-by-zero operation on a given algebra + is its zero map:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J.zero().operator().is_zero() + True + + """ + return self.matrix().is_zero() + + + 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 @@ -432,3 +627,62 @@ 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) + 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 list(zip(eigenvalues, projectors))