totaro@brahms.BERKELEY.EDU (Burt Totaro) (02/12/86)
This note gives a counterexample to a problem proposed by Sam White
on net.math.
The problem was: Let M(n) be the algebra of n x n matrices over a field.
Let V be a linear subspace of M(n) consisting entirely of nilpotent matrices.
(A matrix A is nilpotent if A^m = 0 for some integer m.)
Show that V cannot generate M(n) as a ring.
For the counterexample, take the field to be the real or complex numbers,
and let V be spanned by the following two matrices:
0 0 0 0 1 0
A = 1 0 0 B = 0 0 1
0 -1 0 0 0 0
One finds that any matrix C which is a linear combination of these two
satisfies C^3=0, and hence is nilpotent. However, if you write out
the matrices 1,AB,BA; A,AA,AAB; B,BB,BBA; then you will see that the first
three span the diagonal matrices, the second three span the lower
triangular ones, etc. That is, V generates the ring of 3x3 matrices.
The problem was originally intended to help answer the question:
Given a linear subspace which generates the algebra M(n), can we find
upper and lower bounds for the number of multiplications we need to do
to get the whole algebra? That is, in the above example we needed only
products of at most three elements of the subspace (BBA, for instance.)
As far as I know, this broader question is still wide open.