How to generalize $D$-stability

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.

Short version

References in corpus (1)

• A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?