Summary: A matrix is called sign regular of order k if every minor of order i has the same sign for each i = 1; 2; : : : ; k. At one extreme of the sign regular matrices lies the well studied and important class of totally positive matrices. The purpose here is to initiate a study of the other extreme of sign regular matrices, namely the totally negative matrices (i.e., all minors negative). Many aspects of this class are considered including: existence, spectral properties, inverses, Schur complements, and factorizations.

15A48

minors, sign regular matrices, totally positive, totally negative