Full sign-invertibility and symplectic matrices

Abstract

An n × 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 ∈ Q( H), A −1 exists and A −1 ∈ Q( H −1). If, in addition, H −1 is sign-invertible [implying ( H −1) −1 = H], H is said to be fully sign-invertible and ( H, H −1) is called a sign-invertible pair. Given an n × n sign pattern H, a symplectic pair in Q( H) is a pair of matrices ( A, D) such that A ∈ Q( H), D ∈ Q( H), and A T D = I. (Symplectic pairs are a pattern generalization of orthogonal matrices which arise from a special symplectic matrix found in n-body problems in celestial mechanics [1].) We 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. We proceed to develop sufficient conditions for H's full sign-invertibility 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.