Principally Unimodular Skew-Symmetric Matrices

A square matrix is principally unimodular if every principal submatrix has determinant 0 or 1. Let A be a symmetric (0; 1)-matrix, with a zero diagonal. A PU-orientation of A is a skew-symmetric signing of A that is PU. If A 0 is a PU-orientation of A, then, by a certain decomposition of A, we can construct every PU-orientation of A from A 0. This construction is based on the fact that the PU-orientations of indecomposable matrices are unique up to negation and multiplication of certain rows and corresponding columns by ?1. This generalizes the well-known result of Camion, that if a (0; 1)-matrix can be signed to be totally unimodular then the signing is unique up to multiplying certain rows and columns by ?1. Camion's result is an easy but crucial step in proving Tutte's famous excluded minor characterization of totally unimodular matrices.