Full sign-invertibility and symplectic matrices

Abstract

AbstractAn 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 ATD = 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