X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=5944c0779a8b7a63b1fa41897947cef4dbee83bb;hb=5cbb93016e4b192d2a2d7be81014a55a33c9a8f9;hp=85d45715dae0d787e622293e4f4320d755536911;hpb=259d256fb765350eb6691efe1765c9f4e2a121bd;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 85d4571..5944c07 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -1,3 +1,5 @@
+from itertools import izip
+
 from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
@@ -78,7 +80,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         elif n == 1:
             return self
         else:
-            return (self.operator()**(n-1))(self)
+            return (self**(n-1))*self
 
 
     def apply_univariate_polynomial(self, p):
@@ -458,6 +460,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ...
             ValueError: element is not invertible
 
+        Proposition II.2.3 in Faraut and Koranyi 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::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: x = J.random_element()
+            sage: (not x.operator().is_invertible()) or (
+            ....:    x.operator().inverse()(J.one()) == x.inverse() )
+            True
+
         """
         if not self.is_invertible():
             raise ValueError("element is not invertible")
@@ -754,7 +767,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: n_max = RealSymmetricEJA._max_test_case_size()
             sage: n = ZZ.random_element(1, n_max)
             sage: J1 = RealSymmetricEJA(n,QQ)
-            sage: J2 = RealSymmetricEJA(n,QQ,False)
+            sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
             sage: X = random_matrix(QQ,n)
             sage: X = X*X.transpose()
             sage: x1 = J1(X)
@@ -830,7 +843,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         """
         B = self.parent().natural_basis()
         W = self.parent().natural_basis_space()
-        return W.linear_combination(zip(B,self.to_vector()))
+        return W.linear_combination(izip(B,self.to_vector()))
 
 
     def norm(self):
@@ -968,10 +981,10 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: not x.is_invertible() or (
             ....:   x.quadratic_representation(x.inverse())*Qx
             ....:   ==
-            ....:   2*x.operator()*Qex - Qx )
+            ....:   2*Lx*Qex - Qx )
             True
 
-            sage: 2*x.operator()*Qex - Qx == Lxx
+            sage: 2*Lx*Qex - Qx == Lxx
             True
 
         Property 5:
@@ -1009,7 +1022,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
 
 
-    def subalgebra_generated_by(self):
+    def subalgebra_generated_by(self, orthonormalize_basis=False):
         """
         Return the associative subalgebra of the parent EJA generated
         by this element.
@@ -1048,7 +1061,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             0
 
         """
-        return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
+        return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
 
 
     def subalgebra_idempotent(self):
@@ -1160,8 +1173,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         TESTS:
 
-        The trace inner product is commutative, bilinear, and satisfies
-        the Jordan axiom:
+        The trace inner product is commutative, bilinear, and associative::
 
             sage: set_random_seed()
             sage: J = random_eja()
@@ -1181,7 +1193,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ....:              a*x.trace_inner_product(z) )
             sage: actual == expected
             True
-            sage: # jordan axiom
+            sage: # associative
             sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
             True