Characterization of star sign patterns that require HnHn

The refined inertia of a square real matrix B  , denoted ri(B)ri(B), is the ordered 4-tuple (n+(B),n−(B),nz(B),2np(B)), where n+(B)n+(B) (resp., n−(B)n−(B)) is the number of eigenvalues of B   with positive (resp., negative) real part, nz(B)nz(B) is the number of zero eigenvalues of B  , and 2np(B)2np(B) is the number of pure imaginary eigenvalues of B  . For n≥3n≥3, the set of refined inertias Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)}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. An n×nn×n sign pattern AA is said to require HnHn if Hn={ri(B)|B∈Q(A)}Hn={ri(B)|B∈Q(A)}. The star sign patterns of order n≥5n≥5 that require HnHn are characterized. More specifically, it is shown that for each n≥5n≥5, a star sign pattern requires HnHn if and only if it is equivalent to one of the five sign patterns identified in the paper.