A New Proof of Hopf's Inequality Using a Complex Extension of the Hilbert Metric

2019/06/12 by Wendi Han, Han, Wendi, Guangyue Han +1
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Graph theory and applications #Information Theory (cs.IT) #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1906.04875

openalex publication_date 2019/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known from the Perron-Frobenius theory that the spectral gap of a positive square matrix is positive. In this paper, we give a more quantitative characterization of the spectral gap. More specifically, using a complex extension of the Hilbert metric, we show that the so-called spectral ratio of a positive square matrix is upper bounded by its Birkhoff contraction coefficient, which in turn yields a lower bound on its spectral gap.

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