Finding a similar matrix

Question

Let two square matrices $A$ and $P$ be given. How to find an invertible matrix $P_1$ so that $P_1^{-1} A P_1 = PA$?

Answer 1

In linear algebra, two $~n\times n~$ matrices $~A~$ and $~P~$ are called similar if there exists an invertible $~n\times n~$ matrix $~P_1~$ such that $$P=P_1^{-1}AP_1$$

So the equality given in the question i.e., $~P_1^{-1} A P_1 = PA~$ is only valid for $~A=I_{n \times n}~$.