If we know nilpotent matrix $N\neq 0$, and $PN\neq 0$ for some idempotent matrix $P$, with the property $PN=NP$, can $P$ be unique determined?

Question

Just as the title:If we know nilpotent matrix $N\neq 0$, and $PN\neq 0$ for some idempotent matrix $P$, with the property $PN=NP$, can $P$ be unique determined?

(nilpotent means $N^k=0$ for some $k$, idempotent means $P^2=P$)

Answer 1

No. Take$$N=\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix},\ P_1=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix},$$and $P_2=\operatorname{Id}$. Then $N$ is nilpotent, $N\neq0$, $P_1$ and $P_2$ are idempotent, $NP_1=P_1N\neq0$, and $NP_2=P_2N\neq0$.