Some bounds for determinants of relatively $D$-stable matrices

TL;DR

This paper investigates relatively D-stable matrices, providing new bounds for their determinants, conditions for stability, and spectral sector estimates, with applications to diagonally stable and dominant matrices.

Contribution

It generalizes the Hadamard inequality for relatively D-stable matrices and estimates spectral sector gaps for specific D-stable matrix classes.

Findings

Derived upper bounds for determinants of relatively D-stable matrices.

Established sufficient conditions for relative D-stability.

Estimated spectral sector gaps for certain D-stable matrices.

Abstract

In this paper, we study the class of relatively $D$-stable matrices and provide the conditions, sufficient for relative $D$-stability. We generalize the well-known Hadamard inequality, to provide upper bounds for the determinants of relatively $D$-stable and relatively additive $D$-stable matrices. For some classes of $D$-stable matrices, we estimate the sector gap between matrix spectra and the imaginary axis. We apply the developed technique to obtain upper bounds for determinants of some classes of $D$-stable matrices, e.g. diagonally stable, diagonally dominant and matrices with $Q^2$-scalings.