Overview
--------
This is a collection of design notes that should eventually wind up in
the documentation. I'm just not sure where they go yet.

Matrix representations
----------------------

All algebras have a "matrix representation" of their elements. This is
the original, ambient representation of the elements as either column
(n-by-1) or square (n-by-n) matrices. For example, the elements of the
Jordan spin algebra are column vectors, and the elements of real
symmetric matrix algebra are... real symmetric matrices.

The CombinatorialFreeModule class actually supports such an
alternative representation of its generators since it subclasses
IndexedGenerators. However, using matrices as the index set turns out
to be ugly: printing the generators, and especially printing an
algebra element as a sum of said generators comes out wonky, since
the matrices require more than one line. For example,

    sage: A = matrix(QQ,[[1,2],[3,4]])
    sage: B = matrix(QQ,[[5,6],[7,8]])
    sage: A.set_immutable()
    sage: B.set_immutable()
    sage: M = CombinatorialFreeModule(QQ,[A,B],bracket=False,prefix="")
    sage: 2*M(A) + 3*M(B)
    2*[1 2]
    [3 4] + 3*[5 6]
    [7 8]

And things only get worse if you leave the prefix in there to
distinguish between e.g. the super- and sub-algebra elements
corresponding to the same matrix. Thus, we store out own copy
of the matrix generators, and have our own set of methods for
accessing them.

Why allow matrix representations for all algebras, rather than just
the matrix algebras?

1. We already have a to_vector() operation that turns an algebra
   element into a vector whose coordinates live in the algebra's
   base_ring(). Adding a to_matrix() operation is a natural
   generalization of that.

2. Having access to the ambient coordinates in a general way is useful
   when converting between other coordinate systems. If we have two
   subalgebras B and C of A, we can use to_matrix() to go from, say,
   C -> A -> B rather than having to convert from C to B directly.

3. When constructing a Cartesian product algebra, we don't know a
   priori whether or not the result will have matrix-algebra
   factors. We can figure it out at runtime, but it would be nice if
   DirectSumEJA always returned the same class. Maybe more
   importantly, if a Cartesian product has one matrix and one
   non-matrix factor, then what would its own matrix representation
   look like? We want to delegate to the factors...


Basis normalization
-------------------
For performance reasons, we prefer the algebra constructors to
orthonormalize their own bases. We _could_ ask the user to do that,
but there's a good reason to do it ourselves: if _we_ run
Gram-Schmidt, then we can compute/store the matrix that undoes the
process. Undoing the change-of-coordinates allows us to perform some
computations in the original basis (like the "characteristic
polynomial of"), which can be much faster when the original basis
contains only rational entries.

Debugging
---------
There are several places in the code where we set check_field=False
and check_axioms=False because the theory guarantees that they will be
satisfied. Well, you know what they say about theory and practice. The
first thing you should do when a problem is discovered it replace all
of those with check_field=True and check_axioms=True, and then re-run
the test suite.