I'm having a difficutly proving the following:

Let $A,B \in M\_{n\times n} (\mathbb{R})$ be similar matrices. Let $P \in M\_{n\times n} (\mathbb{R})$ be an invertible matrix satisfying $P^{-1}AP = B.$ Let $v$ be an eigenvector of $A$ corresponding to the eigenvalue $\lambda$. Prove that $P^{-1}v$ is an eigenvector of $B$ corresponding to the eigenvalue $\lambda$.

I've tried looking the the common characteristic polynomial, but didn't get anywhere.