Let $M \in M_{n}(\mathbb{C})$. Show that $M$ is diagonalizable if and only if for every polynomial $P(X) \in \mathbb{C}[X]$ such that $P(M)$ is nilpotent, $P(M) = 0$.
Let $M \in M_{n}(\mathbb{C})$. Show that $M$ is diagonalizable if and only if for every polynomial $P(X) \in \mathbb{C}[X]$ such that $P(M)$ is nilpotent, $P(M) = 0$.