eja: update todo, and rename "natural" to "matrix".

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

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 739bff334c5aa2069dbf09244e7a7fa39ceaccb3..6547668965a690b883a5ac59757ef1e625016604 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -2,11 +2,6 @@ from sage.matrix.constructor import matrix
 from sage.modules.free_module import VectorSpace
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 
-# TODO: make this unnecessary somehow.
-from sage.misc.lazy_import import lazy_import
-lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
-lazy_import('mjo.eja.eja_element_subalgebra',
-            'FiniteDimensionalEuclideanJordanElementSubalgebra')
 from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
 from mjo.eja.eja_utils import _mat2vec
 
@@ -507,6 +502,14 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         if not self.is_invertible():
             raise ValueError("element is not invertible")
 
+        if self.parent()._charpoly_coefficients.is_in_cache():
+            # We can invert using our charpoly if it will be fast to
+            # compute. If the coefficients are cached, our rank had
+            # better be too!
+            r = self.parent().rank()
+            a = self.characteristic_polynomial().coefficients(sparse=False)
+            return (-1)**(r+1)*sum(a[i+1]*self**i for i in range(r))/self.det()
+
         return (~self.quadratic_representation())(self)
 
 
@@ -968,15 +971,16 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
 
 
-    def natural_representation(self):
+    def to_matrix(self):
         """
-        Return a more-natural representation of this element.
+        Return an (often more natural) representation of this element as a
+        matrix.
 
-        Every finite-dimensional Euclidean Jordan Algebra is a
-        direct sum of five simple algebras, four of which comprise
-        Hermitian matrices. This method returns the original
-        "natural" representation of this element as a Hermitian
-        matrix, if it has one. If not, you get the usual representation.
+        Every finite-dimensional Euclidean Jordan Algebra is a direct
+        sum of five simple algebras, four of which comprise Hermitian
+        matrices. This method returns a "natural" matrix
+        representation of this element as either a Hermitian matrix or
+        column vector.
 
         SETUP::
 
@@ -988,7 +992,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: J = ComplexHermitianEJA(3)
             sage: J.one()
             e0 + e3 + e8
-            sage: J.one().natural_representation()
+            sage: J.one().to_matrix()
             [1 0 0 0 0 0]
             [0 1 0 0 0 0]
             [0 0 1 0 0 0]
@@ -1001,7 +1005,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             sage: J = QuaternionHermitianEJA(3)
             sage: J.one()
             e0 + e5 + e14
-            sage: J.one().natural_representation()
+            sage: J.one().to_matrix()
             [1 0 0 0 0 0 0 0 0 0 0 0]
             [0 1 0 0 0 0 0 0 0 0 0 0]
             [0 0 1 0 0 0 0 0 0 0 0 0]
@@ -1014,15 +1018,14 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             [0 0 0 0 0 0 0 0 0 1 0 0]
             [0 0 0 0 0 0 0 0 0 0 1 0]
             [0 0 0 0 0 0 0 0 0 0 0 1]
-
         """
-        B = self.parent().natural_basis()
-        W = self.parent().natural_basis_space()
+        B = self.parent().matrix_basis()
+        W = self.parent().matrix_space()
 
         # This is just a manual "from_vector()", but of course
         # matrix spaces aren't vector spaces in sage, so they
         # don't have a from_vector() method.
-        return W.linear_combination(zip(B,self.to_vector()))
+        return W.linear_combination( zip(B, self.to_vector()) )
 
 
     def norm(self):
@@ -1323,6 +1326,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         """
+        from mjo.eja.eja_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
         return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis)