Consider any matrix $A \in \mathbb R^{n \times n}$ with the p-norm ||A||_{p} < 1.
- I would like to show that $\lim\limits_{k \rightarrow \infty}{A^k} = 0$.
- Consider the reverted scenario. Let $\lim\limits_{k \rightarrow \infty}{A^k} = 0$,
I actually have two questions regarding number 1. It's not stated which p-norm, does this mean it has to be valid for all p-norms? And then I'm a little lost on to how to prove this. Does this mean all elements of the matrix are smaller than 1?
For number 2 I have a hunch it's false. Consider a matrix with $1$ on the top right corner, $A^k$ will be $0$ but it's $p=1$ norm is 1. (and therefore also all other norms?)
Many thanks in advance!