A bound on the joint spectral radius using the diagonals

Abstract

The primary aim of this paper is to establish bounds on the joint spectral radius for a finite set of nonnegative matrices based on their diagonal elements. The efficacy of this approach is evaluated in comparison to existing and related results in the field. In particular, let $\Sigma$ be any finite set of $D\times D$ nonnegative matrices with the largest value $U$ and the smallest value $V$ over all positive entries. For each $i=1,\dots,D$, let $m_i$ be any number so that there exist $A_1,\dots,A_{m_i}\in\Sigma$ satisfying $(A_1\dots A_{m_i})_{i,i} > 0$, or let $m_i=1$ if there are no such matrices. We prove that the joint spectral radius $\rho(\Sigma)$ is bounded by \[ \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}} \le \rho(\Sigma) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}}. \]