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 \(i=1,\ldots ,D\), let \(m_i\) be any number so that there exist \(A_1,\ldots ,A_{m_i}\in \Sigma \) satisfying \((A_1\ldots A_{m_i})_{i,i} > 0\), or let \(m_i=1\) if there are no such matrices. We prove that the joint spectral radius \(\rho (\Sigma )\) is bounded by

$$\begin{aligned} \begin{aligned}&\max _i \root m_i \of {\max _{A_1,\ldots ,A_{m_i}\in \Sigma } (A_1\ldots A_{m_i})_{i,i}} \le \rho (\Sigma ) \\&\quad \le \max _i \root m_i \of {\left( \frac{UD}{V}\right) ^{3D^2} \max _{A_1,\ldots ,A_{m_i}\in \Sigma } (A_1\ldots A_{m_i})_{i,i}}. \end{aligned} \end{aligned}$$