Full Sign-Invertibility and Symplectic Matrices

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 2 Q(H), A 1 exists and A 1 2 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 2 Q(H);D 2 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-inverseH 1, and use this to cast a necessary condition for full sign-invertibility of H. We proceed to develop su cient conditions forH'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.