X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=2eb267727235946ed94681d973c915407c5d3dc1;hb=ce96a6f610c1c2a6837bacd893fed164d2085a03;hp=ee33e4ba89fcc3c47be1f9467042dae5e678c859;hpb=08aba469c5f8d8947a543f8882fa676ed165e7ee;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index ee33e4b..2eb2677 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -430,6 +430,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.inverse() == J.from_vector(x_inverse) True + Trying to invert a non-invertible element throws an error: + + sage: JordanSpinEJA(3).zero().inverse() + Traceback (most recent call last): + ... + ValueError: element is not invertible + TESTS: The identity element is its own inverse:: @@ -455,14 +462,6 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: (not x.is_invertible()) or (x.inverse().inverse() == x) True - The zero element is never invertible:: - - sage: set_random_seed() - sage: J = random_eja().zero().inverse() - Traceback (most recent call last): - ... - ValueError: element is not invertible - Proposition II.2.3 in Faraut and Korányi says that the inverse of an element is the inverse of its left-multiplication operator applied to the algebra's identity, when that inverse exists:: @@ -574,10 +573,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: - The identity element is never nilpotent:: + The identity element is never nilpotent, except in a trivial EJA:: sage: set_random_seed() - sage: random_eja().one().is_nilpotent() + sage: J = random_eja() + sage: J.one().is_nilpotent() and not J.is_trivial() False The additive identity is always nilpotent:: @@ -621,11 +621,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: The zero element should never be regular, unless the parent - algebra has dimension one:: + algebra has dimension less than or equal to one:: sage: set_random_seed() sage: J = random_eja() - sage: J.dimension() == 1 or not J.zero().is_regular() + sage: J.dimension() <= 1 or not J.zero().is_regular() True The unit element isn't regular unless the algebra happens to @@ -633,7 +633,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = random_eja() - sage: J.dimension() == 1 or not J.one().is_regular() + sage: J.dimension() <= 1 or not J.one().is_regular() True """ @@ -677,14 +677,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: - The zero and unit elements are both of degree one:: + The zero and unit elements are both of degree one in nontrivial + algebras:: sage: set_random_seed() sage: J = random_eja() - sage: J.zero().degree() - 1 - sage: J.one().degree() - 1 + sage: d = J.zero().degree() + sage: (J.is_trivial() and d == 0) or d == 1 + True + sage: d = J.one().degree() + sage: (J.is_trivial() and d == 0) or d == 1 + True Our implementation agrees with the definition:: @@ -1040,6 +1043,69 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): + def spectral_decomposition(self): + """ + Return the unique spectral decomposition of this element. + + ALGORITHM: + + Following Faraut and Korányi's Theorem III.1.1, we restrict this + element's left-multiplication-by operator to the subalgebra it + generates. We then compute the spectral decomposition of that + operator, and the spectral projectors we get back must be the + left-multiplication-by operators for the idempotents we + seek. Thus applying them to the identity element gives us those + idempotents. + + Since the eigenvalues are required to be distinct, we take + the spectral decomposition of the zero element to be zero + times the identity element of the algebra (which is idempotent, + obviously). + + SETUP:: + + sage: from mjo.eja.eja_algebra import RealSymmetricEJA + + EXAMPLES: + + The spectral decomposition of the identity is ``1`` times itself, + and the spectral decomposition of zero is ``0`` times the identity:: + + sage: J = RealSymmetricEJA(3,AA) + sage: J.one() + e0 + e2 + e5 + sage: J.one().spectral_decomposition() + [(1, e0 + e2 + e5)] + sage: J.zero().spectral_decomposition() + [(0, e0 + e2 + e5)] + + TESTS:: + + sage: J = RealSymmetricEJA(4,AA) + sage: x = sum(J.gens()) + sage: sd = x.spectral_decomposition() + sage: l0 = sd[0][0] + sage: l1 = sd[1][0] + sage: c0 = sd[0][1] + sage: c1 = sd[1][1] + sage: c0.inner_product(c1) == 0 + True + sage: c0.is_idempotent() + True + sage: c1.is_idempotent() + True + sage: c0 + c1 == J.one() + True + sage: l0*c0 + l1*c1 == x + True + + """ + P = self.parent() + A = self.subalgebra_generated_by(orthonormalize_basis=True) + result = [] + for (evalue, proj) in A(self).operator().spectral_decomposition(): + result.append( (evalue, proj(A.one()).superalgebra_element()) ) + return result def subalgebra_generated_by(self, orthonormalize_basis=False): """ @@ -1077,13 +1143,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: A(x^2) == A(x)*A(x) True - By definition, the subalgebra generated by the zero element is the - one-dimensional algebra generated by the identity element:: + By definition, the subalgebra generated by the zero element is + the one-dimensional algebra generated by the identity + element... unless the original algebra was trivial, in which + case the subalgebra is trivial too:: sage: set_random_seed() sage: A = random_eja().zero().subalgebra_generated_by() - sage: A.dimension() - 1 + sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1 + True """ return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)