Sign patterns that require or allow particular refined inertias (Q448371)

This paper deals with sign patterns of matrices that require or allow particular refined inertias.NEWLINENEWLINEThe refined inertia \(ri(A)\) of an \(n \times n\) real matrix is an ordered 4-tuple \((n_+, n_-, n_z, 2n_p)\) of nonnegative integers summing to \(n\), where \(n_+\) is the number of eigenvalues of \(A\) with positive real part, \(n_-\) is the number of eigenvalues of \(A\) with negative real part, \(n_z\) is the number of zero eigenvalues of \(A\) and \(2n_p\) is the number of nonzero pure imaginary eigenvalues of \(A\). On the other hand, the authors consider \(n \times n\) sign pattern matrices, where a sign pattern \(S=[s_{ij}]\) is a matrix with entries in \(\{+,-,0 \}\). Its associated sign pattern class of matrices is \(Q(S)=\{ A=[a_{ij}] \;: \;\text{sign}(a_{ij})=s_{ij} \;\text{for all} \;i,j \}\).NEWLINENEWLINEThe authors focus their attention on three particular refined inertias for a sign pattern \(H_n=\{(0,n,0,0), (0,n-2,0,2), (2,n-2,0,0) \}\). A sign pattern \(S_n\) requires refined inertia \(H_n\) if \(H_n=ri(S_n)\), and \(S_n\) allows refined inertia \(H_n\) if \(H_n \subseteq ri(S_n)\).NEWLINENEWLINEThe authors analyze sign patterns that require or allow \(H_n\). Firstly, they study sign patterns of order two, three and four, and afterwards, they investigate sign patterns with negative main diagonal of any order.NEWLINENEWLINEFinally, three examples from different applications to dynamical systems are given relating the previous concepts to Hopf bifurcation in dynamical systems.