Are all matrices composed of squares of smaller matrices of some nilpotency index $k$, nilpotent with index $k$?

Question

Are all matrices composed of squares of smaller matrices of some nilpotency index $k$, nilpotent with index $k$?

E.g. $\def\b{\begin{bmatrix}}\def\e{\end{bmatrix}}$

$$N_K=\b N_k&0&\dots\\0&N_k&0&\dots\\0&0&N_k&0&\dots\\\dots&\dots&\dots&\dots&\dots\\0&\dots&&0&N_k\e$$

I have tested a few cases, and I haven't found a counter example, but I can't see why it would be true. Any ideas?

Answer 1

Hint

If $A$ is a block matrix as you described

$$A=\b B_1&0&\dots\\0&B_2&0&\dots\\0&0&B_3&0&\dots\\\dots&\dots&\dots&\dots&\dots\\0&\dots&&0&B_p\e$$

then $$A^n=\b B_1^n&0&\dots\\0&B_2^n&0&\dots\\0&0&B_3^n&0&\dots\\\dots&\dots&\dots&\dots&\dots\\0&\dots&&0&B_p^n\e$$ Can you take it from here?