eja: don't pointlessly orthonormalize in subalgebra_idempotent().

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

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index e1ea609efce334a9800ce162ba74d1342662da41..235047a153f0594bf450987ae2988a5873d92f36 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -795,7 +795,23 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         ALGORITHM:
 
-        .........
+        First we handle the special cases where the algebra is
+        trivial, this element is zero, or the dimension of the algebra
+        is one and this element is not zero. With those out of the
+        way, we may assume that ``self`` is nonzero, the algebra is
+        nontrivial, and that the dimension of the algebra is at least
+        two.
+
+        Beginning with the algebra's unit element (power zero), we add
+        successive (basis representations of) powers of this element
+        to a matrix, row-reducing at each step. After row-reducing, we
+        check the rank of the matrix. If adding a row and row-reducing
+        does not increase the rank of the matrix at any point, the row
+        we've just added lives in the span of the previous ones; thus
+        the corresponding power of ``self`` lives in the span of its
+        lesser powers. When that happens, the degree of the minimal
+        polynomial is the rank of the matrix; if it never happens, the
+        degree must be the dimension of the entire space.
 
         SETUP::
 
@@ -838,7 +854,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
             sage: x = random_eja().random_element()
             sage: x.degree() == x.minimal_polynomial().degree()
             True
-
         """
         n = self.parent().dimension()
 
@@ -1440,7 +1455,10 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         if self.is_nilpotent():
             raise ValueError("this only works with non-nilpotent elements!")
 
-        J = self.subalgebra_generated_by()
+        # The subalgebra is transient (we return an element of the
+        # superalgebra, i.e. this algebra) so why bother
+        # orthonormalizing?
+        J = self.subalgebra_generated_by(orthonormalize=False)
         u = J(self)
 
         # The image of the matrix of left-u^m-multiplication
@@ -1461,14 +1479,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         # subspace... or do we? Can't we just solve, knowing that
         # A(c) = u^(s+1) should have a solution in the big space,
         # too?
-        #
-        # Beware, solve_right() means that we're using COLUMN vectors.
-        # Our FiniteDimensionalAlgebraElement superclass uses rows.
         u_next = u**(s+1)
         A = u_next.operator().matrix()
         c = J.from_vector(A.solve_right(u_next.to_vector()))
 
-        # Now c is the idempotent we want, but it still lives in the subalgebra.
+        # Now c is the idempotent we want, but it still lives in
+        # the subalgebra.
         return c.superalgebra_element()