I am trying to find the matrices $ M \in M_n (\mathbb{R})$ such that $M$ is similar to $M^2$.
I tried to use the fact that if these matrices are similar, then they have the same eigenvalues, but I could not really get anywhere with this.
I would be grateful if you could at least give me a hint. Thank you.
To prove that $A\sim A^k$, we only need to prove that $J\sim J^k$,where J denotes any Jordan block of the Jordan canonical form of A.
Also,$J=\lambda I+J_0$,the characteristic polynomial of $J$ is $(\lambda-a)^{r_i}$.Through discussion with determinant factors and minimal polynomial,we know that the Jordan canonical form of $J_{r_i}(\lambda_i)^m$ is $J_{r_i}(\lambda_i^m)$,so we need nonzero eigenvalues to be $1$(when $k =2$).
Plus, when we look the Jordan canonical form of $J_{r_i}(0)^m$, we can prove that the result should be $diag${ ${J_{q}(0)}$ , ${J_{q}(0)}$,$\cdots$} So the answer is obvious.