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

Wendi Han; Guangyue Han

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arXiv · 1906.04875

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

Wendi Han·Guangyue Han

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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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Wendi Han, Guangyue Han. 2019-06-12. A New Proof of Hopf's Inequality Using a Complex Extension of the Hilbert Metric. https://arxiv.org/abs/1906.04875

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