Full sign-invertibility and symplectic matrices
From authors' abstract: ``An \(n\times n\) sign pattern \(H\) is said to be sign-invertible if there exists a sign pattern \(H^{-1}\) (called the sign inverse of \(H\)) such that, for all matrices \(A\in Q(H)\) (the sign class of \(H\)), \(A^{- 1}\) exists and \(A^{-1}\in Q(H^{- 1})\). If, in addition, \(H^{-1}\) is sign-invertible, \(H\) is said to be fully sign-invertible and \((H, H^{- 1})\) is called a sign-invertible pair. Given an \(n\times n\) sign pattern \(H\), a symplectic pair in \(Q(H)\) is a pair of matrices \((A, D)\) such that \(A\in Q(H)\), \(D\in Q(H)\), and \(A^TD= I\).'' The authors discuss the digraphical relationship between a sign-invertible pattern \(H\) and its sign inverse \(H^{ -1}\), and use this to cast a necessary condition for full sign-invertibility of \(H\). They proceed to develop sufficient conditions for full sign-invertibility of \(H\) in terms of allowed paths and cycles in the digraph of \(H\), and conclude with a complete characterization of those sign patterns that require symplectic pairs.
scientific article; zbMATH DE number 841756