Two classes of symmetric sign patterns that require unique inertia

Abstract

A sign pattern is a matrix whose entries are from the set {+,−,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A is a set i(A)={i(B)=(i +(B),i −(B),i 0(B)) | B=B T H∈Q(A)}, where i + (B) (respectively,i − (B),i 0(B) denotes the number of positive (respectively, negative, zero) eigenvalues. That the symmetric sign pattern A requires unique intertia means i(B 1)=i(B 2) for all real symmetric matrices B 1,B 2∈Q(A). The purpose of this paper is to characterize double star and cycle sign patterns that require unique inertia. Further, their unique inertia is also obtained.