Sign patterns that require [formula omitted] exist for each n ≥ 4

Abstract

The refined inertia of a square real matrix A is the ordered 4-tuple (n+,n−,nz,2np), where n+ (resp., n−) is the number of eigenvalues of A with positive (resp., negative) real part, nz is the number of zero eigenvalues of A, and 2np is the number of nonzero pure imaginary eigenvalues of A. For n≥3, the set of refined inertias Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)} is important for the onset of Hopf bifurcation in dynamical systems. We say that an n×n sign pattern A requires Hn if Hn={ri(B)|B∈Q(A)}. Bodine et al. conjectured that no n×n irreducible sign pattern that requires Hn exists for n sufficiently large, possibly n≥8. However, for each n≥4, we identify three n×n irreducible sign patterns that require Hn, which resolves this conjecture.