eja: support conversion of naturally-represented elements into an EJA.

This lets us convert e.g. a symmetric 3x3 matrix to an EJA element
without having to mess with the basis/coordinates ourselves.

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index e30c34128b564499d0485c3d0f9fcb68a826e6c7..b25839815289ba6f60630a14b790dbfe29f8969b 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -160,6 +160,51 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         An element of a Euclidean Jordan algebra.
         """
 
+        def __init__(self, A, elt=None):
+            """
+            EXAMPLES:
+
+            The identity in `S^n` is converted to the identity in the EJA::
+
+                sage: J = RealSymmetricSimpleEJA(3)
+                sage: I = identity_matrix(QQ,3)
+                sage: J(I) == J.one()
+                True
+
+            This skew-symmetric matrix can't be represented in the EJA::
+
+                sage: J = RealSymmetricSimpleEJA(3)
+                sage: A = matrix(QQ,3, lambda i,j: i-j)
+                sage: J(A)
+                Traceback (most recent call last):
+                ...
+                ArithmeticError: vector is not in free module
+
+            """
+            # Goal: if we're given a matrix, and if it lives in our
+            # parent algebra's "natural ambient space," convert it
+            # into an algebra element.
+            #
+            # The catch is, we make a recursive call after converting
+            # the given matrix into a vector that lives in the algebra.
+            # This we need to try the parent class initializer first,
+            # to avoid recursing forever if we're given something that
+            # already fits into the algebra, but also happens to live
+            # in the parent's "natural ambient space" (this happens with
+            # vectors in R^n).
+            try:
+                FiniteDimensionalAlgebraElement.__init__(self, A, elt)
+            except ValueError:
+                natural_basis = A.natural_basis()
+                if elt in natural_basis[0].matrix_space():
+                    # Thanks for nothing! Matrix spaces aren't vector
+                    # spaces in Sage, so we have to figure out its
+                    # natural-basis coordinates ourselves.
+                    V = VectorSpace(elt.base_ring(), elt.nrows()**2)
+                    W = V.span( _mat2vec(s) for s in natural_basis )
+                    coords =  W.coordinates(_mat2vec(elt))
+                    FiniteDimensionalAlgebraElement.__init__(self, A, coords)
+
         def __pow__(self, n):
             """
             Return ``self`` raised to the power ``n``.