eja: reword two existing tests.

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 4947fe82bb981401a09a039306516b4b0631113a..4acf513d3a8aa6b498621af5908b11363f77d420 100644 (file)
--- 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)