How to show $f(A)$ is diagonalizable?

Question

If $A$ is diagonalizable and $f(·)$ is a polynomial, how to show that $f(A)$ is diagonalizable?

Answer 1

HINT

We have that

$$A=SDS^{-1} \implies A^2=SDS^{-1}SDS^{-1}=SD^2S^{-1} \implies \dots A^n=SD^nS^{-1}$$

Answer 2

Try to show (based on gimusi's hint) that if $A= BDB^{-1}$, then

$$f(A)= Bf(D)B^{-1}$$ as well..

and a polynomial applied to the subgroup of diagonal matrices stays in there.