Q: If complex matrices $\rm{A}$ and $\rm{B}$ satisfy $\rm{rank}\ (\rm{AB}-\rm{BA})\ \le1$, then there exists a revertible matrix $\rm{P}$ such that $\rm{P^{-1}AP}$ and $\rm{P^{-1}BP}$ are both upper triangular matrices.
I have figured out the proof when the condition is restricted to $\rm{AB-BA} = 0$. Since each characteristic subspace of $\rm{A}$ is also an invariant subspace of $\rm{B}$, one can readily see that $\rm{A}$ and $\rm{B}$ share a common eigenvector, and the conclusion can be drawn by induction. However, when it comes to $\rm{rank\ (AB-BA)= 1}$, I find it hard to find a common eigenvector of $\rm{A}$, $\rm{B}$.
Am I right in this way of thinking? Is there any other elegant proof? Thanks for any help!
