CONVEX SETS OF STABLE MATRICES

Abstract

A complex matrix is positive (negative) stable if all its eigenvalues lie in the open right (left) half-plane. Stable matrices play a central role in the study of systems of differential equations. In this thesis, we will study the stability of the product and Kronecker product of two matrices. We will explore the principal pivot transform and the Cayley transform of a stable matrix and state sufficient conditions that ensure the stability of these transforms. In addition, we obtain a result about the stability of rank-one updated matrices. Furthermore, we shall discuss the stability of convex hulls of stable matrices. Finally, we present some results about the existence of a positive eigenvalue based on the concept of pointed cones, then we will use these results to study stable matrices, P−matrices, and copositive matrices.