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

## TL;DR

This paper introduces a novel approach using a complex extension of the Hilbert metric to provide a quantitative bound on the spectral gap of positive matrices, enhancing understanding of their spectral properties.

## Contribution

It presents a new proof of Hopf's inequality by linking the spectral ratio to the Birkhoff contraction coefficient through a complex Hilbert metric extension.

## Key findings

- Spectral ratio is bounded by the Birkhoff contraction coefficient.
- Provides a lower bound on the spectral gap of positive matrices.
- Introduces a complex extension of the Hilbert metric for spectral analysis.

## 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.