X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_operator.py;h=689b7ecec26c09a8423ca440ee6b00429d50073b;hb=HEAD;hp=ee33dbf53b36fd9851d31169d5699041f460328e;hpb=99ca9f8c24194ad6be7b8e325575e58b53429c2b;p=sage.d.git diff --git a/mjo/eja/eja_operator.py b/mjo/eja/eja_operator.py index ee33dbf..689b7ec 100644 --- a/mjo/eja/eja_operator.py +++ b/mjo/eja/eja_operator.py @@ -2,13 +2,47 @@ from sage.matrix.constructor import matrix from sage.categories.all import FreeModules from sage.categories.map import Map -class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): +class EJAOperator(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 EJAOperator + + 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: EJAOperator(V,J,M) + Traceback (most recent call last): + ... + TypeError: domain must be a finite-dimensional Euclidean + Jordan algebra + sage: EJAOperator(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 EJA + + # I guess we should check this, because otherwise you could + # pass in pretty much anything algebraish. + if not isinstance(domain_eja, EJA): + raise TypeError('domain must be a finite-dimensional ' + 'Euclidean Jordan algebra') + if not isinstance(codomain_eja, EJA): + raise TypeError('codomain must be a finite-dimensional ' + 'Euclidean Jordan algebra') F = domain_eja.base_ring() if not (F == codomain_eja.base_ring()): @@ -27,7 +61,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): # The Map initializer will set our parent to a homset, which # is explicitly NOT what we want, because these ain't algebra # homomorphisms. - super(FiniteDimensionalEuclideanJordanAlgebraOperator,self).__init__(parent) + super().__init__(parent) # Keep a matrix around to do all of the real work. It would # be nice if we could use a VectorSpaceMorphism instead, but @@ -42,7 +76,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import JordanSpinEJA EXAMPLES:: @@ -50,7 +84,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: J = JordanSpinEJA(3) 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 = EJAOperator(J,J,id) sage: f(x) == x True @@ -64,7 +98,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import ( ....: JordanSpinEJA, ....: RealSymmetricEJA ) @@ -75,8 +109,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: J = RealSymmetricEJA(2) sage: id = identity_matrix(J.base_ring(), J.dimension()) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) - sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) + sage: f = EJAOperator(J,J,id) + sage: g = EJAOperator(J,J,id) sage: f + g Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: @@ -93,15 +127,15 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: id1 = identity_matrix(J1.base_ring(), 3) sage: J2 = JordanSpinEJA(3) sage: id2 = identity_matrix(J2.base_ring(), 3) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J1,id1) - sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,J2,id2) + sage: f = EJAOperator(J1,J1,id1) + sage: g = EJAOperator(J2,J2,id2) sage: f + g Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for +: ... """ - return FiniteDimensionalEuclideanJordanAlgebraOperator( + return EJAOperator( self.domain(), self.codomain(), self.matrix() + other.matrix()) @@ -114,7 +148,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import ( ....: JordanSpinEJA, ....: HadamardEJA, @@ -125,26 +159,22 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: J1 = JordanSpinEJA(3) 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: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J1, - ....: J2, - ....: mat1) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J2, - ....: J3, - ....: mat2) + sage: mat2 = matrix(AA, [[7,8]]) + sage: g = EJAOperator(J1, J2, mat1) + sage: f = EJAOperator(J2, J3, mat2) sage: f*g Linear operator between finite-dimensional Euclidean Jordan 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( + return EJAOperator( other.domain(), self.codomain(), self.matrix()*other.matrix()) @@ -166,14 +196,14 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: id = identity_matrix(J.base_ring(), J.dimension()) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) + sage: f = EJAOperator(J,J,id) sage: ~f Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: @@ -184,7 +214,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): Codomain: Euclidean Jordan algebra of dimension 3 over... """ - return FiniteDimensionalEuclideanJordanAlgebraOperator( + return EJAOperator( self.codomain(), self.domain(), ~self.matrix()) @@ -201,7 +231,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES: @@ -209,8 +239,8 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): We can scale an operator on a rational algebra by a rational number:: sage: J = RealSymmetricEJA(2) - sage: e0,e1,e2 = J.gens() - sage: x = 2*e0 + 4*e1 + 16*e2 + sage: b0,b1,b2 = J.gens() + sage: x = 2*b0 + 4*b1 + 16*b2 sage: x.operator() Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: @@ -231,7 +261,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): """ try: if other in self.codomain().base_ring(): - return FiniteDimensionalEuclideanJordanAlgebraOperator( + return EJAOperator( self.domain(), self.codomain(), self.matrix()*other) @@ -242,8 +272,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): # This should eventually delegate to _composition_ after performing # some sanity checks for us. - mor = super(FiniteDimensionalEuclideanJordanAlgebraOperator,self) - return mor.__mul__(other) + return super().__mul__(other) def _neg_(self): @@ -252,14 +281,14 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: id = identity_matrix(J.base_ring(), J.dimension()) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) + sage: f = EJAOperator(J,J,id) sage: -f Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: @@ -270,7 +299,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): Codomain: Euclidean Jordan algebra of dimension 3 over... """ - return FiniteDimensionalEuclideanJordanAlgebraOperator( + return EJAOperator( self.domain(), self.codomain(), -self.matrix()) @@ -282,7 +311,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA TESTS: @@ -292,7 +321,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): sage: J = RealSymmetricEJA(2) sage: id = identity_matrix(J.base_ring(), J.dimension()) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) + sage: f = EJAOperator(J,J,id) sage: f^0 + f^1 + f^2 Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: @@ -314,7 +343,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): else: mat = self.matrix()**n - return FiniteDimensionalEuclideanJordanAlgebraOperator( + return EJAOperator( self.domain(), self.codomain(), mat) @@ -328,22 +357,22 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import JordanSpinEJA EXAMPLES:: sage: J = JordanSpinEJA(2) sage: id = identity_matrix(J.base_ring(), J.dimension()) - sage: FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) + sage: EJAOperator(J,J,id) Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: [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 " @@ -362,14 +391,14 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: id = identity_matrix(J.base_ring(),J.dimension()) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id) + sage: f = EJAOperator(J,J,id) sage: f - (f*2) Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: @@ -383,6 +412,72 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): 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 EJAOperator + 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 = EJAOperator(J1,J2,M) + sage: L.is_zero() + True + sage: M = matrix(R, [ [0, 0], + ....: [0, 1], + ....: [0, 0] ]) + sage: L = EJAOperator(J1,J2,M) + sage: L.is_zero() + False + + TESTS: + + The left-multiplication-by-zero operation on a given algebra + is its zero map:: + + 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. @@ -414,7 +509,6 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): The identity operator is its own inverse:: - sage: set_random_seed() sage: J = random_eja() sage: idJ = J.one().operator() sage: idJ.inverse() == idJ @@ -422,7 +516,6 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): 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) @@ -465,14 +558,12 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): 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 @@ -488,14 +579,14 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: sage: J = RealSymmetricEJA(2) sage: mat = matrix(J.base_ring(), J.dimension(), range(9)) - sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,mat) + sage: f = EJAOperator(J,J,mat) sage: f.matrix() [0 1 2] [3 4 5] @@ -512,7 +603,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): SETUP:: - sage: from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator + sage: from mjo.eja.eja_operator import EJAOperator sage: from mjo.eja.eja_algebra import RealSymmetricEJA EXAMPLES:: @@ -542,9 +633,9 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): EXAMPLES:: - sage: J = RealSymmetricEJA(4,AA) + sage: J = RealSymmetricEJA(4) sage: x = sum(J.gens()) - sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) + sage: A = x.subalgebra_generated_by() sage: L0x = A(x).operator() sage: sd = L0x.spectral_decomposition() sage: l0 = sd[0][0] @@ -577,7 +668,7 @@ class FiniteDimensionalEuclideanJordanAlgebraOperator(Map): # for the spectral theorem to work. us[i] = us[i]/us[i].norm() mat = us[i].column()*us[i].row() - Pi = FiniteDimensionalEuclideanJordanAlgebraOperator( + Pi = EJAOperator( self.domain(), self.codomain(), mat)