On asymptotic properties of matrix semigroups with an invariant cone

TL;DR

This paper investigates the asymptotic behavior of matrix semigroups with invariant cones, establishing continuity of the joint spectral subradius and convergence of spectral radius and trace for products of matrices sharing an invariant cone.

Contribution

It proves the continuity of the joint spectral subradius near sets of matrices with invariant cones and demonstrates convergence of spectral radius and trace for such matrices under primitivity.

Findings

Joint spectral subradius is continuous near matrices with invariant cones.

Spectral radius and trace of matrix products converge to the joint spectral radius.

Results apply to matrices sharing an invariant cone with one being primitive.

Abstract

Recently, several research efforts showed that the analysis of joint spectral characteristics of sets of matrices is greatly eased when these matrices share an invariant cone. In this short note we prove two new results in this direction. We prove that the joint spectral subradius is continuous in the neighborhood of sets of matrices that leave an embedded pair of cones invariant. We show that the (averaged) maximal spectral radius, as well as the maximal trace, of products of length t, converge towards the joint spectral radius when the matrices share an invariant cone, and addi- tionally one of them is primitive.