X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=e7dff7529026cf007056a5ecd66a8f9fba98562a;hb=6b80adf4c1b41455d6e2afe003cdbc96452d0928;hp=5944c0779a8b7a63b1fa41897947cef4dbee83bb;hpb=5cbb93016e4b192d2a2d7be81014a55a33c9a8f9;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 5944c07..e7dff75 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -1,3 +1,5 @@
+# -*- coding: utf-8 -*-
+
 from itertools import izip
 
 from sage.matrix.constructor import matrix
@@ -34,7 +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,
@@ -375,7 +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()
@@ -405,7 +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:
@@ -460,7 +463,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             ...
             ValueError: element is not invertible
 
-        Proposition II.2.3 in Faraut and Koranyi says that the inverse
+        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::
 
@@ -471,6 +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")