How to generalize $D$-stability

TL;DR

This paper introduces a unified framework for various matrix stability concepts by defining $({\mathfrak D},{\mathcal G},\circ)$-stability, surveys existing results, and explores properties and relations among different stability classes.

Contribution

It generalizes multiple known stability types into a single framework and analyzes their properties and interrelations.

Findings

Unified approach to matrix stability concepts.

Survey of existing results and open problems.

Analysis of properties and relations among stability classes.

Abstract

In this paper, we introduce the following concept which generalizes known definitions of multiplicative and additive $D$-stability, Schur $D$-stability, $H$-stability, $D$-hyperbolicity and many others. Given a subset ${\mathfrak D} \subset {\mathbb C}$, a matrix class ${\mathcal G} \subset {\mathcal M}^{n \times n}$ and a binary operation $\circ$ on ${\mathcal M}^{n \times n}$, an $n \times n$ matrix $\mathbf A$ is called $({\mathfrak D}, {\mathcal G}, \circ)$-stable if $\sigma({\mathbf G}\circ {\mathbf A}) \subset {\mathfrak D}$ for any ${\mathbf G} \in {\mathcal G}$. Such an approach allows us to unite several well-known matrix problems and to consider common ways of their analysis. Here, we make a survey of existing results and open problems on different types of stability, study basic properties of $({\mathfrak D}, {\mathcal G}, \circ)$-stable matrices and relations between different $({\mathfrak D}, {\mathcal G}, \circ)$-stability classes.