An example of non-similar matrices with same eigenvalues, rank and determinant

Question

I have found the following example in Horn's and Johnson's Matrix Analysis book. Let $$A=\begin{bmatrix} 0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0 \end{bmatrix}$$ $$B=\begin{bmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0 \end{bmatrix}$$ The two matrices have the same eigenvalues (characteristic polynomial), trace, determinant and rank. But since $A^2=0$ and $B^2\neq0$, we can conclude that they are not similar. Can you explain me the last claim i.e. why it implies that they cannot be similar? Please cite relevant theorems or lemmas.

Answer 1

Suppose $A^2=0$ and $B^2 \neq 0$ but $B=P^{-1}AP$

then $B^2=P^{-1}A^2P=0$ which is a contradiction.