Suppose that $A$ and $B$ are linear transformations (on the same finite dimensional vector space) such that $A^2=A$ and $B^2=B$. Is it true that $A$ and $B$ are similar if and only if rank($A$)=rank($B$)?
As far as I can tell, it is true. What I have for the forward direction is this:
$A=C^{-1}BC$
$CA=BC$
And by a theorem, we can say, rank($CA$)=rank($BC$). Further, by the same theorem, we can say, rank($A$)=rank($B$) because $C$ is invertible.
I'm stuck on the reverse direction though. I can put an invertible $C$ in there, but past that, I don't think I can make the assumption that $CA=BC$.
Thanks.