If we have any two nilpotent $3\times3$ matrices $A, B$ with rank $2$, are they always similar?
I started by considering their nullity, which is $1$. Then, since they are nilpotent, their only eigenvalue is $0$, thus their eigenspaces are their nullspaces.
Since their nullity is $1$, then their JCF must only have one jordan block, which means their JCF is the same (a $3\times3$ jordan block with $0$ as the diagonal), thus they are similar.
My reasoning may be flawed, please give me some insight.