From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 13 Oct 2019 00:15:09 +0000 (-0400)
Subject: eja: add a test for the inverse via charpoly.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=e9845713afb8ed88273d2b8dfe170ca8f11a5290;p=sage.d.git

eja: add a test for the inverse via charpoly.
---

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")