X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=e7dff7529026cf007056a5ecd66a8f9fba98562a;hb=e9845713afb8ed88273d2b8dfe170ca8f11a5290;hp=eee8f69bd76ddfba49e3cb4531f55d0e970ebd1d;hpb=bbd6d4c6e39870b0936949b510e70af2b5358f9e;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index eee8f69..e7dff75 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -407,7 +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:
@@ -473,6 +474,22 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ....:    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")