If $A\in M_n(\mathbb{C})$ is a nilpotent matrix then $A$ is similar to $2A$ I am trying to prove this property but the truth is I cannot find how to express the matrix $ P $ such that $$ A=P^{-1}2AP \Leftrightarrow PA =2AP$$
I tried a diagonal matrix but it didn't get me anywhere, can someone help me?