X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=2bfa3714420eff322b6b2fb19e177107a191ef29;hb=b5363a907ef2d36ca2912e18b16edfcc65abe6cd;hp=4947fe82bb981401a09a039306516b4b0631113a;hpb=f911ca107b6ad2a92547a255311ef7f16978feac;p=sage.d.git

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 4947fe8..2bfa371 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)
@@ -1355,7 +1359,7 @@ class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
             p = z**2 - 2
             if p.is_irreducible():
                 field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
-            S = [ s.change_ring(field) for s in S ]
+                S = [ s.change_ring(field) for s in S ]
             self._basis_normalizers = tuple(
                 ~(self.__class__.natural_inner_product(s,s).sqrt())
                 for s in S )
@@ -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)
@@ -1455,7 +1463,7 @@ class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
             p = z**2 - 2
             if p.is_irreducible():
                 field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
-            S = [ s.change_ring(field) for s in S ]
+                S = [ s.change_ring(field) for s in S ]
             self._basis_normalizers = tuple(
                 ~(self.__class__.natural_inner_product(s,s).sqrt())
                 for s in S )