Add another TODO for later.

[sage.d.git] / mjo / eja / eja_algebra.py

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 698aa37e2d6698c680ec51d3db1b272e16b6df4d..2bfa3714420eff322b6b2fb19e177107a191ef29 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -941,7 +941,7 @@ def _complex_hermitian_basis(n, field):
 
 
 
-def _quaternion_hermitian_basis(n, field, normalize):
+def _quaternion_hermitian_basis(n, field):
     """
     Returns a basis for the space of quaternion Hermitian n-by-n matrices.
 
@@ -959,7 +959,7 @@ def _quaternion_hermitian_basis(n, field, normalize):
 
         sage: set_random_seed()
         sage: n = ZZ.random_element(1,5)
-        sage: B = _quaternion_hermitian_basis(n, QQ, False)
+        sage: B = _quaternion_hermitian_basis(n, QQ)
         sage: all( M.is_symmetric() for M in B )
         True
 
@@ -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 )