Matrices having a positive determinant and all other minors nonpositive

Abstract

The class of square matrices of order n having a positive determinant and all their minors up to order n−1 nonpositive is considered. A characterization of these matrices based on the Cauchon algorithm is presented, which provides an easy test for their recognition. Furthermore, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too. For both results, we require that the entry in the bottom-right position is negative.