If I have that a matrix $A \in \mathbb{R}^{n \times n}$ is centered and m-banded i.e.
$\textit{m-banded:}$
there is an index $l$ such that $$a_{i,j}=0 \hspace{4mm} if \hspace{2mm}j \notin[i-l,i-l+m]$$
$\textit{centered}$: $$m=2s, \hspace{4mm}s \in \mathbb{Z}, \hspace{3mm}l=\frac{m}{2}$$
how to prove that $A^k$ is still centered and $km-$banded?