I'm looking for a way to define quadratic matrices $M\in \mathbb{R}^{m \times m}$ such that when I raise them to the power of $n$ they won't have any entry growing towards infinity. (They also shouldn't converge to zero if possible, but that's not so important.)
Is there some criterion assuring this property (but still not restricting the possible values too much)?
(I know, this is school math, but I forgot and I couldn't find the solution anywhere on the internet.)
(So far I looked at the standard recursive formula for each entry of $M^n$: $$M^n_{ik}=\sum_{j=1}^m M_{ij} \cdot M^{n-1}_{jk}$$ yielding a nested chain of sums and multiplications, but I was not able to resolve it in any effective way.)