Some bounds for determinants of relatively $D$-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.