X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=d787c5fc1366411fe6f6a3b549d8dbd285037d8b;hb=c66155061d8b520f7c90a7e866c0f9366202d64c;hp=c47156d31e2c6989e1fe863091794520f395523c;hpb=02c754829b2f2e8378561e6afd7cbfab2577f3f4;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index c47156d..d787c5f 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -165,6 +165,21 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.apply_univariate_polynomial(p) 0 + The characteristic polynomials of the zero and unit elements + should be what we think they are in a subalgebra, too:: + + sage: J = RealCartesianProductEJA(3) + sage: p1 = J.one().characteristic_polynomial() + sage: q1 = J.zero().characteristic_polynomial() + sage: e0,e1,e2 = J.gens() + sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3 + sage: p2 = A.one().characteristic_polynomial() + sage: q2 = A.zero().characteristic_polynomial() + sage: p1 == p2 + True + sage: q1 == q2 + True + """ p = self.parent().characteristic_polynomial() return p(*self.to_vector()) @@ -368,6 +383,16 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.is_invertible() == (x.det() != 0) True + Ensure that the determinant is multiplicative on an associative + subalgebra as in Faraut and Koranyi's Proposition II.2.2:: + + sage: set_random_seed() + sage: J = random_eja().random_element().subalgebra_generated_by() + sage: x = J.random_element() + sage: y = J.random_element() + sage: (x*y).det() == x.det()*y.det() + True + """ P = self.parent() r = P.rank() @@ -684,6 +709,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA, ....: random_eja) TESTS: @@ -733,6 +759,20 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.apply_univariate_polynomial(p) 0 + The minimal polynomial is invariant under a change of basis, + and in particular, a re-scaling of the basis:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,5) + sage: J1 = RealSymmetricEJA(n) + sage: J2 = RealSymmetricEJA(n,QQ,False) + sage: X = random_matrix(QQ,n) + sage: X = X*X.transpose() + sage: x1 = J1(X) + sage: x2 = J2(X) + sage: x1.minimal_polynomial() == x2.minimal_polynomial() + True + """ if self.is_zero(): # We would generate a zero-dimensional subalgebra @@ -800,10 +840,37 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): """ B = self.parent().natural_basis() - W = B[0].matrix_space() + W = self.parent().natural_basis_space() return W.linear_combination(zip(B,self.to_vector())) + def norm(self): + """ + The norm of this element with respect to :meth:`inner_product`. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealCartesianProductEJA) + + EXAMPLES:: + + sage: J = RealCartesianProductEJA(2) + sage: x = sum(J.gens()) + sage: x.norm() + sqrt(2) + + :: + + sage: J = JordanSpinEJA(4) + sage: x = sum(J.gens()) + sage: x.norm() + 2 + + """ + return self.inner_product(self).sqrt() + + def operator(self): """ Return the left-multiplication-by-this-element @@ -887,7 +954,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): Property 2 (multiply on the right for :trac:`28272`): - sage: alpha = QQ.random_element() + sage: alpha = J.base_ring().random_element() sage: (alpha*x).quadratic_representation() == Qx*(alpha^2) True @@ -992,7 +1059,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: A = random_eja().zero().subalgebra_generated_by() sage: A - Euclidean Jordan algebra of dimension 0 over Rational Field + Euclidean Jordan algebra of dimension 0 over... sage: A.one() 0 @@ -1109,22 +1176,19 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: - The trace inner product is commutative:: + The trace inner product is commutative, bilinear, and satisfies + the Jordan axiom: sage: set_random_seed() sage: J = random_eja() - sage: x = J.random_element(); y = J.random_element() - sage: x.trace_inner_product(y) == y.trace_inner_product(x) - True - - The trace inner product is bilinear:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() + sage: x = J.random_element(); sage: y = J.random_element() sage: z = J.random_element() - sage: a = QQ.random_element(); + sage: # commutative + sage: x.trace_inner_product(y) == y.trace_inner_product(x) + True + sage: # bilinear + sage: a = J.base_ring().random_element(); sage: actual = (a*(x+z)).trace_inner_product(y) sage: expected = ( a*x.trace_inner_product(y) + ....: a*z.trace_inner_product(y) ) @@ -1135,15 +1199,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): ....: a*x.trace_inner_product(z) ) sage: actual == expected True - - The trace inner product satisfies the compatibility - condition in the definition of a Euclidean Jordan algebra:: - - sage: set_random_seed() - sage: J = random_eja() - sage: x = J.random_element() - sage: y = J.random_element() - sage: z = J.random_element() + sage: # jordan axiom sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z) True @@ -1152,3 +1208,30 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): raise TypeError("'other' must live in the same algebra") return (self*other).trace() + + + def trace_norm(self): + """ + The norm of this element with respect to :meth:`trace_inner_product`. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealCartesianProductEJA) + + EXAMPLES:: + + sage: J = RealCartesianProductEJA(2) + sage: x = sum(J.gens()) + sage: x.trace_norm() + sqrt(2) + + :: + + sage: J = JordanSpinEJA(4) + sage: x = sum(J.gens()) + sage: x.trace_norm() + 2*sqrt(2) + + """ + return self.trace_inner_product(self).sqrt()