Sign properties of Metzler matrices with applications

TL;DR

This paper investigates the sign stability properties of Metzler matrices, providing characterizations, conditions, and generalizations, with applications to jump Markov processes and matrix convex hulls.

Contribution

It introduces new criteria for sign-stability of Metzler matrices, including block and mixed types, and extends the concept to Ker$_+(B)$-sign-stability with practical applications.

Findings

Sign-stability characterized by Hurwitz stability and graph acyclicity.

Conditions for sign-stability of convex hulls of Metzler matrices.

Sufficient conditions for Ker$_+(B)$-sign-stability using sign-matrix inverses.

Abstract

Several results about sign properties of Metzler matrices are obtained. It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms the negativity of the diagonal elements of the Metzler sign-matrix and the acyclicity of the associated directed graph. Similar results are obtained for the case of Metzler block-matrices and Metzler mixed-matrices, the latter being a class of Metzler matrices containing both sign- and real-type entries. The problem of assessing the sign-stability of the convex hull of a finite and summable family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of common Lyapunov functions for all the matrices in the convex hull is obtained. The concept of sign-stability is then generalized to the concept of Ker$_+(B)$-sign-stability, a problem that arises in the analysis of certain jump Markov processes. A sufficient condition for the Ker$_+(B)$-sign-stability of Metzler sign-matrices is obtained and formulated using inverses of sign-matrices and the concept of $L^+$-matrices. Several applications of the results are discussed in the last section.