Matrices Having A Positive Determinant And All Other Minors Nonpositive

Abstract

The class of square real matrices of order n having a positive determinant and all other minors up to order n − 1 nonpositive are called sign regular matrices with signature (−1, . . . , −1, 1). In this thesis, such matrices are introduced and a characterization of them is presented which provides an easy test for their recognition based on the so-called the Cauchon Algorithm. The value of the entry (2, 2) of the matrix resulting upon application of the Cauchon algorithm to such a sign regular matrix plays a fundamental role in our characterization. Therefore, the possible values of the entry (2, 2) are explored. Finally, it is shown that all matrices lying between two matrices of this class with respect to the so-called checkerboard ordering are contained in this class, too.