eja: allow matrix algebras of "size" zero.

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

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index fc64510dde701f7f820456d9972691fe07f71178..04929467d114dbb23cc4fcc15f11c80df58addf0 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -688,6 +688,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
                         for k in range(n) )
 
         L_x = matrix(F, n, n, L_x_i_j)
+
+        r = None
+        if self.rank.is_in_cache():
+            r = self.rank()
+            # There's no need to pad the system with redundant
+            # columns if we *know* they'll be redundant.
+            n = r
+
         # Compute an extra power in case the rank is equal to
         # the dimension (otherwise, we would stop at x^(r-1)).
         x_powers = [ (L_x**k)*self.one().to_vector()
@@ -696,7 +704,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         AE = A.extended_echelon_form()
         E = AE[:,n:]
         A_rref = AE[:,:n]
-        r = A_rref.rank()
+        if r is None:
+            r = A_rref.rank()
         b = x_powers[r]
 
         # The theory says that only the first "r" coefficients are
@@ -984,7 +993,24 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
             J = MatrixEuclideanJordanAlgebra(QQ,
                                              basis,
                                              normalize_basis=False)
-            return J._charpoly_coefficients()
+            a = J._charpoly_coefficients()
+
+            # Unfortunately, changing the basis does change the
+            # coefficients of the characteristic polynomial, but since
+            # these are really the coefficients of the "characteristic
+            # polynomial of" function, everything is still nice and
+            # unevaluated. It's therefore "obvious" how scaling the
+            # basis affects the coordinate variables X1, X2, et
+            # cetera. Scaling the first basis vector up by "n" adds a
+            # factor of 1/n into every "X1" term, for example. So here
+            # we simply undo the basis_normalizer scaling that we
+            # performed earlier.
+            #
+            # TODO: make this access safe.
+            XS = a[0].variables()
+            subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i]
+                          for i in range(len(XS)) }
+            return tuple( a_i.subs(subs_dict) for a_i in a )
 
 
     @staticmethod
@@ -1002,6 +1028,9 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
         # is supposed to hold the entire long vector, and the subspace W
         # of V will be spanned by the vectors that arise from symmetric
         # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
+        if len(basis) == 0:
+            return []
+
         field = basis[0].base_ring()
         dimension = basis[0].nrows()
 
@@ -1159,6 +1188,11 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra):
         sage: x.operator().matrix().is_symmetric()
         True
 
+    We can construct the (trivial) algebra of rank zero::
+
+        sage: RealSymmetricEJA(0)
+        Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
     """
     @classmethod
     def _denormalized_basis(cls, n, field):
@@ -1432,6 +1466,11 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra):
         sage: x.operator().matrix().is_symmetric()
         True
 
+    We can construct the (trivial) algebra of rank zero::
+
+        sage: ComplexHermitianEJA(0)
+        Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
     """
 
     @classmethod
@@ -1727,6 +1766,11 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra):
         sage: x.operator().matrix().is_symmetric()
         True
 
+    We can construct the (trivial) algebra of rank zero::
+
+        sage: QuaternionHermitianEJA(0)
+        Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
+
     """
     @classmethod
     def _denormalized_basis(cls, n, field):