Ok. Things like this make me feel like the original papers should be more easily available.Uh, okay. I think we understand each other, though. So there's not too much more to be said here. I'll give you Ashcroft and Mermin's statement of Bloch's Theorem (emphasis added):
"The eigenstates of the one-electron Hamiltonian [tex]H=-\hbar^2\nabla^2/2m+U(\vec r)[/tex], where [tex]U(\vec r)=U(\vec r + \vec R)[/tex] for all [tex]\vec R[/tex] in a Bravais lattive, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice."
But, again, you don't have to make that choice. cheers.
There's one more thing! I got the idea of the proof with matrices, but are you sure this result generalizes to the infinite dimensional spaces? Looks like, that forming linear combinations of some given eigenvectors to form new ones becomes a more delicate issue then.