I'm confused. Taking a (admittedly quick) look at the problem it would appear to me that this is merely a question about if we have a degenerate eigenvalue can we diagonalize the matrix?

If we have a matrix with two eigenvalues (each with degenerate eigenspaces to be general) we can always put the matrix in block diagonal form. So the question should become whether we can diagonalize an n x n matrix that has an n degenerate eigenvalue.

Please let me know if I'm over-simplifying things and missed something.

-Dan