From a26c6d48d496531069c0f89842e60c4407873314 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 23 Aug 2019 10:42:54 -0400 Subject: [PATCH] eja: reword two existing tests. --- mjo/eja/eja_algebra.py | 20 ++++++++++++++------ 1 file changed, 14 insertions(+), 6 deletions(-) diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 4947fe8..4acf513 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -1325,7 +1325,8 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: (x*y).inner_product(z) == y.inner_product(x*z) True - Our basis is normalized with respect to the natural inner product:: + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) @@ -1333,8 +1334,11 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: all( b.norm() == 1 for b in J.gens() ) True - Left-multiplication operators are symmetric because they satisfy - the Jordan axiom:: + Since our natural basis is normalized with respect to the natural + inner product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the EJA + the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) @@ -1425,7 +1429,8 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: (x*y).inner_product(z) == y.inner_product(x*z) True - Our basis is normalized with respect to the natural inner product:: + Our natural basis is normalized with respect to the natural inner + product unless we specify otherwise:: sage: set_random_seed() sage: n = ZZ.random_element(1,4) @@ -1433,8 +1438,11 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra): sage: all( b.norm() == 1 for b in J.gens() ) True - Left-multiplication operators are symmetric because they satisfy - the Jordan axiom:: + Since our natural basis is normalized with respect to the natural + inner product, and since we know that this algebra is an EJA, any + left-multiplication operator's matrix will be symmetric because + natural->EJA basis representation is an isometry and within the EJA + the operator is self-adjoint by the Jordan axiom:: sage: set_random_seed() sage: n = ZZ.random_element(1,5) -- 2.43.2