Perturbing eigenvalues of non-negative matrices

2014/02/05 by Chi-Kwong Li, Yiu-Tung Poon, Li, Chi-Kwong +4
Computer Science · Mathematics · #15A18 #15A48 #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Point processes and geometric inequalities #Spectral Theory (math.SP) #math.SP #msc:15A18 #msc:15A48

paper · pdf · doi:10.48550/arxiv.1402.0917

17 pages

arxiv created 2014/02/05 · openalex publication_date 2014/02/05 · arxiv updated 2014/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be an irreducible (entrywise) nonnegative n× n matrix with eigenvalues ρ, b+ic,b-ic, λ₄,⋯,λ_n, where ρ is the Perron eigenvalue. It is shown that for any t ∈ [0, ∞) there is a nonnegative matrix with eigenvalues ρ+ t,λ₂+t,λ₃+t, λ₄ ⋯,λ_n, whenever t ≥ γ_n t with γ₃=1, γ₄ = 2, γ₅=√ 5 and γ_n = 2.25 for n ≥ 6. The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an n-sided convex polygon is bounded by γ_n times the maximum area of the triangle lying inside the polygon.

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