Inertially arbitrary tree sign patterns of order 4

Abstract

An n× n sign pattern matrix A is an inertially arbitrary pattern if for every non- negative triple (n1, n2, n3) with n1 + n2 + n3 = n, there is a real matrix in the sign pattern class of A having inertia (n1, n2, n3). An n× n sign pattern matrix A is a spectrally arbitrary pattern if for any given real monic polynomial r(x) of degree n, there is a real matrix in the sign pattern class of A with characteristic polynomial r(x). In this paper, all 4× 4 tree sign pattern matrices that are inertially arbitrary are characterized. As a result, in this paper, it is shown that a 4× 4 tree sign pattern matrix is inertially arbitrary if and only if it is spectrally arbitrary.