X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=e7dff7529026cf007056a5ecd66a8f9fba98562a;hb=e9845713afb8ed88273d2b8dfe170ca8f11a5290;hp=c90893f1f65e92e4032b3b51d4f66d0fc5592900;hpb=02fe7b401586d6ec76028b576b9c61ff6c65d62d;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index c90893f..e7dff75 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -1,3 +1,7 @@
+# -*- coding: utf-8 -*-
+
+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
@@ -32,7 +36,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         Return ``self`` raised to the power ``n``.
 
         Jordan algebras are always power-associative; see for
-        example Faraut and Koranyi, Proposition II.1.2 (ii).
+        example Faraut and Korányi, Proposition II.1.2 (ii).
 
         We have to override this because our superclass uses row
         vectors instead of column vectors! We, on the other hand,
@@ -373,7 +377,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         Ensure that the determinant is multiplicative on an associative
-        subalgebra as in Faraut and Koranyi's Proposition II.2.2::
+        subalgebra as in Faraut and Korányi's Proposition II.2.2::
 
             sage: set_random_seed()
             sage: J = random_eja().random_element().subalgebra_generated_by()
@@ -403,7 +407,8 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+            ....:                                  JordanSpinEJA,
             ....:                                  random_eja)
 
         EXAMPLES:
@@ -458,6 +463,33 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ...
             ValueError: element is not invertible
 
+        Proposition II.2.3 in Faraut and Korányi 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
+
+        Proposition II.2.4 in Faraut and Korányi gives a formula for
+        the inverse based on the characteristic polynomial and the
+        Cayley-Hamilton theorem for Euclidean Jordan algebras::
+
+            sage: set_random_seed()
+            sage: J = ComplexHermitianEJA(3)
+            sage: x = J.random_element()
+            sage: while not x.is_invertible():
+            ....:     x = J.random_element()
+            sage: r = J.rank()
+            sage: a = x.characteristic_polynomial().coefficients(sparse=False)
+            sage: expected  = (-1)^(r+1)/x.det()
+            sage: expected *= sum( a[i+1]*x^i for i in range(r) )
+            sage: x.inverse() == expected
+            True
+
         """
         if not self.is_invertible():
             raise ValueError("element is not invertible")
@@ -830,7 +862,7 @@ 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):
@@ -1009,7 +1041,7 @@ 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 +1080,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             0
 
         """
-        return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
+        return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)
 
 
     def subalgebra_idempotent(self):
@@ -1160,8 +1192,7 @@ 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 +1212,7 @@ 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