A bound on the joint spectral radius using the diagonals

2020/12/01 by Vuong Bui, Bui, Vuong
Computer Science · Engineering · Mathematics · #15A18 #65F15 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2012.00598

openalex publication_date 2020/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Σ be any finite set of D× D nonnegative matrices with the largest value U and the smallest value V over all positive entries. For each i=1,…,D, let m_i be any number so that there exist A₁,…,A_mi∈Σ satisfying (A₁… A_mi)_i,i > 0, or let m_i=1 if there are no such matrices. We prove that the joint spectral radius ρ(Σ) is bounded by max_i √[m_i]max_A₁,…,A_mi∈Σ (A₁… A_mi)_i,i ≤ ρ(Σ) ≤ max_i √[m_i]((UD)/(V))^3D2 max_A₁,…,A_mi∈Σ (A₁… A_mi)_i,i.

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