What is the upper bound of $\operatorname{rank}(\lambda B – BA)$, where $A$ is a $n\times n$ matrix, $B$ is a $m\times n$ matrix and $\lambda$ is an eigenvalue of $A$?
My intuition is that the upper bound is $\operatorname{rank}(\begin{bmatrix} B \\ BA\end{bmatrix})$, but I do not know how to explain it.
Best Answer
Note that $\lambda B - BA = B(\lambda I_n - A)$. It follows that $$ \operatorname{rank}(B - BA) \leq \min\{\operatorname{rank}(B),\operatorname{rank}(\lambda I_n - A)\}. $$ So, your bound is correct in that $$ \operatorname{rank}(B - BA) \leq \operatorname{rank}(B) \leq \operatorname{rank}\pmatrix{B\\ BA}. $$