Matrices with exactly one real positive eigenvalue and the rest having negative or non-positive real parts

Abstract

An $n\times n$ real or complex matrix A is called elliptic if it has exactly one real positive eigenvalue and all of its other eigenvalues have non-positive real parts. The real symmetric elliptic matrices were recently discussed extensively in [4], [16], [19], [24] and have provided many interesting results and applications. However, when the system gets perturbed, the corresponding matrix will no longer be symmetric and such a class of matrices appears in many areas of applied mathematics and sciences. In this paper we study the general real or complex elliptic matrices. We first establish a criterion based on the Hurwitz's sequence of determinants similar to the Routh-Hurwitz's theorem on stable matrices and discuss elliptic matrices from their characteristic polynomials. We then discover that the real or complex elliptic matrices bear close relations with the PN-matrices and the SPN-matrices that appear in the trade theory of economics [3], [9], [11].