Bipartite Graphs and Inverse Sign Patterns of Strong Sign-Nonsingular Matrices

A sign-nonsingular matrix is a square matrix X such that each matrix Y with the same sign pattern as X is nonsingular. If, in addition, the sign pattern of the inverse of Y is the same for all Y, then X is a strong sign-nonsingular matrix. A fully indecomposable matrix is a matrix whose associated bipartite graph is connected and (perfect) matching covered. The bipartite graphs of fully indecomposable, strong sign-nonsingular matrices are characterized and a recursive construction is given. This characterization is used to determine the sign patterns of the inverses of fully indecomposable, strong sign-nonsingular matrices and to develop a recognition algorithm for such sign patterns. Those maximal strong sign-nonsingular matrices whose sign patterns are uniquely determined by the sign patterns of their inverses are also characterized in terms of bipartite graphs.