Eigenvector decomposition on finite fields
For real-valued matrices there are a variety of algorithms that do clever things I don't understand to compute the eigenvalues quickly as limiting sequences.
To experiment: what happens if you run these algorithms with finite fields instead? Seems like you either ought to get convergence or cycles, but it's not clear that either would tell you anything about the eigenvalues.
All this reminds me that I should post my Tubulo solver on Combinatorium, as long as I've got the domain registered... and I still have some work I'd like to do on sliding and match-3 games.