Refined inertias of tree sign-patterns

Abstract

The refined inertia (n+,n ,nz,2np) of a real matrix is the ordered 4-tuple that subdivides the number n0 of eigenvalues with zero real part in the inertia (n+,n ,n0) into those that are exactly zero (nz) and those that are nonzero (2np). For n � 2, 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. Tree sign patterns of order n that require or allow the refined inertias Hn are considered. For n = 4, necessary and sufficient conditions are proved for a tree s pattern (necessarily a path or a star) to require H4. For n � 3, a family of n × n star sign patterns that allows Hn is given, and it is proved that if a star sign pattern requires Hn, then it must have exactly one zero diagonal entry associated with a leaf in its digraph.