A note on sign patterns of order 3 that require particular refined inertias

Abstract

For a real n×n matrix A having n+ (n−) eigenvalues with positive (resp. negative) real part, nz zero eigenvalues and 2np nonzero pure imaginary eigenvalues, the refined inertia of A is ri(A)=(n+,n−,nz,2np). When n=3, let H3={(0,3,0,0),(0,1,0,2),(2,1,0,0)}. A 3×3 sign pattern A requires refined inertia H3 if {ri(A)|A has sign pattern A}=H3. Necessary and sufficient conditions for an irreducible sign pattern to require H3 are given, and used to determine all such sign patterns (up to equivalence). These remove equivalences and complete the list of patterns given in [1, Appendix A].