On the joint spectral radius of nonnegative matrices

Abstract

We give an effective bound of the joint spectral radius $\rho(\Sigma)$ for a finite set $\Sigma$ of nonnegative matrices: For every $n$, \[ \sqrt[n]{\left(\frac{V}{UD}\right)^{D} \max_C \max_{i,j\in C} \max_{A_1,\dots,A_n\in\Sigma}(A_1\dots A_n)_{i,j}} \le \rho(\Sigma) \le \sqrt[n]{D \max_C \max_{i,j\in C} \max_{A_1,\dots,A_n\in\Sigma}(A_1\dots A_n)_{i,j}}, \] where $D\times D$ is the dimension of the matrices, $U,V$ are respectively the largest entry and the smallest entry over all the positive entries of the matrices in $\Sigma$, and $C$ is taken over all strongly connected components in the dependency graph. The dependency graph is a directed graph where the vertices are the dimensions and there is an edge from $i$ to $j$ if and only if $A_{i,j}\ne 0$ for some matrix $A\in\Sigma$. Furthermore, a bound on the norm is also given: If $\rho(\Sigma)>0$ then there exist a nonnegative integer $r$ and two positive numbers $\alpha,\beta$ so that for every $n$, \[ \alpha n^r{\rho(\Sigma)}^n \le \max_{A_1,\dots,A_n\in\Sigma} \|A_1\dots A_n\| \le \beta n^r{\rho(\Sigma)}^n. \] Corollaries of the approach include a simple proof for the joint spectral theorem for finite sets of nonnegative matrices and the convergence rate of some sequences. The method in use is mostly based on Fekete's lemma, for both submultiplicative and supermultiplicative sequences.