I'm asked to prove that if two matrices $A,B$ are similar and $||\cdot||$ is a norm in $\mathbb{C}^n$, then we would have $\lim\_{m\to \infty}||A^m||^{\frac{1}{m}} = \lim\_{m\to \infty}||B^m||^{\frac{1}{m}}$.

What I got:

Since $A,B$ are similar, we have $B= P^{-1}AP$ for some invertible $P$. Then $\lim\_{m\to \infty}||B^m||^{\frac{1}{m}} = \lim\_{m\to \infty}||P^{-1}A^mP||^{\frac{1}{m}}$, but what next?

Thank you