Given a matrix $A \in R^{n \times n}$ with $\rho(A)<1$ (all eigen values strictly inside the unit circle),
how does the sequence $\{||A^k||_{\infty},k \geq 1\}$ converge?
For some matrix $M \in R^{n \times n}$, $||M||_{\infty}$ denotes the usual matrix $\infty$-norm defined as $$||M||_{\infty}=\max_{i=1,\cdots,n} \sum_{j=1}^n |M_{ij}|$$
It is clear that $\lim_{k \to \infty} ||A^k||_{\infty} = 0$. However I am unable to understand how to characterize the convergence to $0$.
Thanks you vey much.