Statement: I can prove that if I apply a function to my matrix (lets call it) "A"........whatever that function does on A, it will do the same thing to the eigenvalues (I can prove this with a similarity transformation I think), so long as the function is basically a linear combination of the powers of "A" or something like that. Question: How do I prove what this function does to the eigen vectors though? Do they remain the same? Do they change? Thanks!!