Full rank factorization in echelon form of totally nonpositive (negative) rectangular matrices

An n×m real matrix A is said to be totally nonpositive (negative) if every minor is nonpositive (negative). In this paper, we study the full rank factorization in echelon form of a totally nonpositive (negative) matrix. This characterization allows us to significantly reduce the number of minors to be checked in order to decide the total negativity of a matrix.