On Generalised Stein's Inequalites and the Inertias of Their Solutions: Application to Pencil Finite Root‐Clustering

Abstract

This note deals with the inertia of the solutions to Generalised Stein's Inequalities (GSI) i.e. Stein's inequalities related to discrete descriptor models described through linear matrix pencils. Only strict inequalities are considered. It is shown that, under some very slight assumptions, there always exists a solution to the GSI and some properties on the inertia of this solution can be derived very easily. As an application, the finite root‐clustering of the pencil associated with a descriptor system in a union of separate discs is considered.