Nearly sign-nonsingular matrices Original Research Article

A real matrix A is nearly sign-nonsingular if every term in the expansion of det A but one has the same sign. We show such matrices can be put into a normal form in which all diagonal entries are negative, all other nonzero entries are positive, and the directed graph of the matrix is intercyclic. With the help of recent results of Metzlar, McCuaig, and Thomassen on intercyclic digraphs, we are able to separate the nearly sign-nonsingular matrices into five classes and to characterize each of these classes. We also obtain two results showing where real matrices having intercyclic digraphs can or cannot be signed in such a way as to belong both to the class of sign-nonsingular matrices and the class of nearly sign-nonsingular matrices.