How to prove that the rank and the nullity of similarity invariants are the same? Given matrix $A$ and $P^{-1}AP$ how do prove that the $\mathrm{rank}(A)$ and $\mathrm{rank}\left(P^{-1}AP\right)$ are the same?
Also, how do you prove that the $\mathrm{nullity}(A)$ and $\mathrm{nullity}\left(P^{-1}AP\right)$ are the same?

$A$ and $P^{-1}AP$ represent the same linear transformation in different bases. Thus, their rank and nullity are equal to the rank and nullity of the linear transformation they both represent. (Strictly speaking, you need to fix bases for both $A$ and $P^{-1}AP$ to speak of "the" linear transformation they represent.
