How to check D-stability: a simple determinantal test

TL;DR

This paper introduces a simple, sufficient determinantal test based on principal minors to check matrix D-stability for matrices of any size, simplifying an otherwise complex problem.

Contribution

It proposes a new, easy-to-verify determinantal criterion for D-stability applicable to matrices of arbitrary size, addressing a longstanding open problem.

Findings

The test is simple to verify and applicable to matrices of any size.

It provides a sufficient condition for D-stability based on principal minors.

The method can be used for analyzing parameter-dependent models.

Abstract

The concept of matrix $D$-stability, introduced in 1958 by Arrow and McManus is of major importance due to the variety of its applications. However, characterization of matrix $D$-stability for dimensions $n > 4$ is considered as a hard open problem. In this paper, we propose a simple way for testing matrix $D$-stability, in terms of the inequalities between principal minors of a matrix. The conditions are just sufficient but they allow to test matrices of an arbitrary size $n$, are easy to verify and can be used for the analysis of parameter-dependent models.