On Semiconvergence, Eventual Positivity, and Accretiveness of Signed Laplacians: A Spectral Abscissa Versus Log-Norm Perspective

• 成果类型: Article
• 署名作者: Zhu, Shouchong; Tian, Ye; Chen, Wei
• 署名单位: Peking University; Peking University
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3488472
• 发表日期: 2025
• 页码: 2622-2629
• 关键词: Laplace equations; Eigenvalues and eigenfunctions; Matrix decomposition; Transmission line matrix methods; vectors; Multi-agent systems; CONVERGENCE; Synthetic aperture sonar; switches; Social networking (online); Eventual exponential positivity; log-norm; quasi-sectorial matrix; signed Laplacian; spectral abscissa

摘要

In this note, we study the relationships among semiconvergence, eventual exponential positivity, and accretiveness of repelling signed Laplacians in a broader context from a spectral abscissa versus logarithmic norm (log-norm) perspective. We give both algebraic and geometrical characterizations of when spectral abscissa equals log-norm for a general real matrix. Furthermore, we characterize the interplay among spectral abscissa, eventual exponential positivity, and log-norm. These results, when applied to the repelling signed Laplacians, expand existing knowledge on relationships among the semiconvergence, eventual exponential positivity, and accretiveness in multiple aspects: first, introduce a new element, i.e., the algebraic connectivity of a signed digraph into the relationship; second, accommodate a relaxed version of accretiveness allowing positive diagonal scaling; third, provide a uniform treatment of repelling signed Laplacian and a generalization of it associated with bipartite consensus.