Stephen: Thanks!

And yeah, I was wondering what the answer was if I don't necessarally demand them to be orthognal, just that I require them to span the space.

Anyways, am right now reading through Down with Determinants. Maybe that'll have the answer in there.

(Actually, the part which I get to, at least for finite dimensional spaces, is already effectively in there: The number of distinct eigenvalues has to equal the dimension of the space. Of course, the question of what has to be true about a linear operator for _that_ to hold is something I'm wondering. :))