X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=9414d2cf27fd2728ef205650b128ab37d1b83949;hb=38ca36bf7be2feb446d99c7cb85ae77e1fb5013c;hp=d459ebe97540abe88f1aef603e29ec994bd1cb74;hpb=aff15cd0c15bd9953531b1d54041f2d90f1d1cff;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index d459ebe..9414d2c 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -5,7 +5,7 @@ are used in optimization, and have some additional nice methods beyond what can be supported in a general Jordan Algebra. """ -from sage.categories.magmatic_algebras import MagmaticAlgebras +from sage.categories.finite_dimensional_algebras_with_basis import FiniteDimensionalAlgebrasWithBasis from sage.structure.element import is_Matrix from sage.structure.category_object import normalize_names @@ -30,16 +30,16 @@ class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMo 2. Inputs and outputs the underlying matrix with respect to COLUMN vectors, unlike the parent class. - 3. Allows us to add, multiply (compose), and invert morphisms in - the obvious way. + 3. Allows us to add, subtract, negate, multiply (compose), and + invert morphisms in the obvious way. If this seems a bit heavyweight, it is. I would have been happy to use a the ring morphism that underlies the finite-dimensional algebra morphism, but they don't seem to be callable on elements of - our EJA, and you can't add/multiply/invert them. + our EJA, and you can't add/multiply/etc. them. """ - def __add__(self, other): + def _add_(self, other): """ Add two EJA morphisms in the obvious way. @@ -86,7 +86,7 @@ class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMo check=False) - def __invert__(self): + def _invert_(self): """ EXAMPLES:: @@ -124,7 +124,7 @@ class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMo return FiniteDimensionalEuclideanJordanAlgebraMorphism(self.parent(), A.inverse()) - def __mul__(self, other): + def _lmul_(self, other): """ Compose two EJA morphisms using multiplicative notation. @@ -162,6 +162,36 @@ class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMo self.matrix()*other.matrix() ) + def _neg_(self): + """ + Negate this morphism. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: x = J.one() + sage: -x.operator() + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational Field + given by matrix + [-1 0 0] + [ 0 -1 0] + [ 0 0 -1] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: -x.operator() in J.Hom(J) + True + + """ + return FiniteDimensionalEuclideanJordanAlgebraMorphism( + self.parent(), + -self.matrix() ) + + def _repr_(self): """ We override only the representation that is shown to the user, @@ -196,6 +226,36 @@ class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMo return "Morphism from {} to {} given by matrix\n{}".format( self.domain(), self.codomain(), self.matrix()) + + def __sub__(self, other): + """ + Subtract one morphism from another using addition and negation. + + EXAMPLES:: + + sage: J = RealSymmetricEJA(2) + sage: L1 = J.one().operator() + sage: L1 - L1 + Morphism from Euclidean Jordan algebra of degree 3 over Rational + Field to Euclidean Jordan algebra of degree 3 over Rational + Field given by matrix + [0 0 0] + [0 0 0] + [0 0 0] + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.operator() - y.operator() in J.Hom(J) + True + + """ + return self + (-other) + + def matrix(self): """ Return the matrix of this morphism with respect to a left-action @@ -222,7 +282,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): raise ValueError("input is not a multiplication table") mult_table = tuple(mult_table) - cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis() + cat = FiniteDimensionalAlgebrasWithBasis(field) cat.or_subcategory(category) if assume_associative: cat = cat.Associative() @@ -1308,7 +1368,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: D = (x0^2 - x_bar.inner_product(x_bar))*D sage: D = D + 2*x_bar.tensor_product(x_bar) sage: Q = block_matrix(2,2,[A,B,C,D]) - sage: Q == x.quadratic_representation() + sage: Q == x.quadratic_representation().operator_matrix() True Test all of the properties from Theorem 11.2 in Alizadeh:: @@ -1317,8 +1377,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() - sage: Lx = x.operator_matrix() - sage: Lxx = (x*x).operator_matrix() + sage: Lx = x.operator() + sage: Lxx = (x*x).operator() sage: Qx = x.quadratic_representation() sage: Qy = y.quadratic_representation() sage: Qxy = x.quadratic_representation(y) @@ -1339,17 +1399,16 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Property 3: - sage: not x.is_invertible() or ( - ....: Qx*x.inverse().vector() == x.vector() ) + sage: not x.is_invertible() or ( Qx(x.inverse()) == x ) True sage: not x.is_invertible() or ( - ....: Qx.inverse() + ....: ~Qx ....: == ....: x.inverse().quadratic_representation() ) True - sage: Qxy*(J.one().vector()) == (x*y).vector() + sage: Qxy(J.one()) == x*y True Property 4: @@ -1362,15 +1421,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: not x.is_invertible() or ( ....: x.quadratic_representation(x.inverse())*Qx ....: == - ....: 2*x.operator_matrix()*Qex - Qx ) + ....: 2*x.operator()*Qex - Qx ) True - sage: 2*x.operator_matrix()*Qex - Qx == Lxx + sage: 2*x.operator()*Qex - Qx == Lxx True Property 5: - sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy + sage: Qy(x).quadratic_representation() == Qy*Qx*Qy True Property 6: @@ -1381,13 +1440,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Property 7: sage: not x.is_invertible() or ( - ....: Qx*x.inverse().operator_matrix() == Lx ) + ....: Qx*x.inverse().operator() == Lx ) True Property 8: sage: not x.operator_commutes_with(y) or ( - ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) ) + ....: Qx(y)^n == Qxn(y^n) ) True """ @@ -1396,9 +1455,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): elif not other in self.parent(): raise TypeError("'other' must live in the same algebra") - L = self.operator_matrix() - M = other.operator_matrix() - return ( L*M + M*L - (self*other).operator_matrix() ) + L = self.operator() + M = other.operator() + return ( L*M + M*L - (self*other).operator() ) def span_of_powers(self):