product of matrices, and its norm

Question

I have a product of matrices $\prod\limits_{i=1}^{n} a_i$, If $b$ is an eigenvalue of $a_i$ for any $i$, then $|b|<1$.

(1) Under what norm or condition, $\|\prod\limits_{i=1}^{n} a_i\|<r<1$

(2) If I had infinite such matrices, $\|\prod\limits_{i=1}^{\infty} a_i\|<r<1$ true for some norm?

(3) What if all the matrices are diagonalizable? In that case, will I get (1) and (2) as desired in some norm?

Thanks for any help.

Answer 1

Perhaps the vector induced norm can work? If $b$ is an eigenvalue of $a_i$ implies that $|b|<1$, then the largest eigenvalue of $a_i$, call it $\tilde{a}_i$ should also be $|\tilde{a}_i|$<1. Taking a look specifically at the statement in the wiki page:

"Any induced operator norm is a submultiplicative matrix norm $\lVert AB \rVert < \lVert A \rVert \lVert B \rVert $ ".

Thus, in the vector induced norm, we should be able to say that \begin{equation} \lVert \prod_{i=1}^{n} a_i \rVert < \prod_{i=1}^{n} \lVert a_i \rVert < \lVert \tilde{a} \rVert^{n}<1, \end{equation} where $\tilde{a}$ is the largest eigenvalue of all the $\tilde{a}_i$'s.

I think this applies for all three points.