eja: do an extra ambient_vector_space() in one method in case its a module.

[sage.d.git] / mjo / eja / euclidean_jordan_algebra.py

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 15ff26ca909cd1fbec4a2f1fc95e69b69f5bd02f..4a2979248fc5e7b90938a0c18b8a7545bb780e97 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -492,8 +492,36 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             We can't use the superclass method because it relies on
             the algebra being associative.
+
+            ALGORITHM:
+
+            The usual way to do this is to check if the determinant is
+            zero, but we need the characteristic polynomial for the
+            determinant. The minimal polynomial is a lot easier to get,
+            so we use Corollary 2 in Chapter V of Koecher to check
+            whether or not the paren't algebra's zero element is a root
+            of this element's minimal polynomial.
+
+            TESTS:
+
+            The identity element is always invertible::
+
+                sage: set_random_seed()
+                sage: J = random_eja()
+                sage: J.one().is_invertible()
+                True
+
+            The zero element is never invertible::
+
+                sage: set_random_seed()
+                sage: J = random_eja()
+                sage: J.zero().is_invertible()
+                False
+
             """
-            return not self.det().is_zero()
+            zero = self.parent().zero()
+            p = self.minimal_polynomial()
+            return not (p(zero) == zero)
 
 
         def is_nilpotent(self):
@@ -598,6 +626,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
         def minimal_polynomial(self):
             """
+            ALGORITHM:
+
+            We restrict ourselves to the associative subalgebra
+            generated by this element, and then return the minimal
+            polynomial of this element's operator matrix (in that
+            subalgebra). This works by Baes Proposition 2.3.16.
+
             EXAMPLES::
 
                 sage: set_random_seed()
@@ -632,25 +667,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 True
 
             """
-            # The element we're going to call "minimal_polynomial()" on.
-            # Either myself, interpreted as an element of a finite-
-            # dimensional algebra, or an element of an associative
-            # subalgebra.
-            elt = None
-
-            if self.parent().is_associative():
-                elt = FiniteDimensionalAlgebraElement(self.parent(), self)
-            else:
-                V = self.span_of_powers()
-                assoc_subalg = self.subalgebra_generated_by()
-                # Mis-design warning: the basis used for span_of_powers()
-                # and subalgebra_generated_by() must be the same, and in
-                # the same order!
-                elt = assoc_subalg(V.coordinates(self.vector()))
-
-            # Recursive call, but should work since elt lives in an
-            # associative algebra.
-            return elt.minimal_polynomial()
+            V = self.span_of_powers()
+            assoc_subalg = self.subalgebra_generated_by()
+            # Mis-design warning: the basis used for span_of_powers()
+            # and subalgebra_generated_by() must be the same, and in
+            # the same order!
+            elt = assoc_subalg(V.coordinates(self.vector()))
+            return elt.operator_matrix().minimal_polynomial()
 
 
         def natural_representation(self):
@@ -853,7 +876,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             # The dimension of the subalgebra can't be greater than
             # the big algebra, so just put everything into a list
             # and let span() get rid of the excess.
-            V = self.vector().parent()
+            #
+            # We do the extra ambient_vector_space() in case we're messing
+            # with polynomials and the direct parent is a module.
+            V = self.vector().parent().ambient_vector_space()
             return V.span( (self**d).vector() for d in xrange(V.dimension()) )