If (A,B) is controllable, is $(A^2,B)$ controllable as well?

Question

I'm assuming a 3x3 matrix with controllability matrix as $[B\ AB\ A^2B]$. I feel that if A is a nilpotent matrix with n=4, then controllability of $(A^2,B)$ would be $[B\ A^2B\ A^4B]=[B\ A^2B \ 0]$ which would make the rank<3 and therefore uncontrollable. Am I right? Or am I missing something?

Answer 1

An obvious example along the lines indicated in your question is the pair $(A,B)$ with $A=\begin{bmatrix}0&1\\0&0\end{bmatrix}$ and $B=\begin{bmatrix}0\\1\end{bmatrix}$. $(A,B)$ is controllable while $(A^2,B)$ is not.

Answer 2

It also depends on the rank of $B$. For example in the extreme case that $B=I$ then even $A=0$ would make $(A,B)$ controllable.