Speed of convergence to the Perron-Frobenius stationary distribution

Abstract

In this paper, we study the second eigenvalue problem for non-negative matrices. Using the Hilbert projective metric, we give a new simple proof of the distance between the first and second eigenvalue whenever the spectral radius is a single peripheral eigenvalue. This article focuses on the convergence speed of (finite-dimensional) linear dynamical systems to the Perron-Frobenius stationary distribution. We extend these results to primitive matrices and cooperative ordinary differential equations, and our proof also extends to some non-autonomous discrete-time systems.