  Wiedemann's paper, introducing his algorithm for sparse and structured matrix
computations over arbitrary fields, also presented a pair of matrix
preconditioners for computations over small fields. The analysis of the second
of these is extended in order to provide more explicit statements of the
expected number of nonzero entries in the matrices obtained as well as bounds
on the probability that such matrices have maximal rank.
  This is part of ongoing work to establish that this matrix preconditioner can
also be used to bound the number of nontrivial nilpotent blocks in the Jordan
normal form of a preconditioned matrix, in such a way that one can also sample
uniformly from the null space of the originally given matrix. If successful
this will result in a black box algorithm for the type of matrix computation
required when using the number field sieve for integer factorization that is
provably reliable and - by a small factor - asymptotically more efficient than
alternative techniques that make use of other matrix preconditioners or require
computations over field extensions.
