A bound on the joint spectral radius using the diagonals

TL;DR

This paper introduces a new method to bound the joint spectral radius of a set of nonnegative matrices using their diagonal elements, providing explicit bounds that improve understanding of matrix behavior.

Contribution

The paper presents novel bounds on the joint spectral radius based solely on diagonal entries, offering a new approach compared to existing spectral radius estimation techniques.

Findings

Derived explicit bounds for the joint spectral radius using diagonal elements.

Compared the new bounds with existing results, demonstrating efficacy.

Provided theoretical guarantees for the bounds in terms of matrix entries.

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 \[ \max_i \sqrt[m_i]{\max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}} \le \rho(\Sigma) \le \max_i \sqrt[m_i]{\left(\frac{UD}{V}\right)^{3D^2} \max_{A_1,\dots,A_{m_i}\in\Sigma} (A_1\dots A_{m_i})_{i,i}}. \]