Characterizations of matrices with nonzero signed row compound

Abstract

A real square matrix A is called a sign-nonsingular (SNS) matrix if every matrix with the same sign pattern as A is not singular. An m×n matrix A with term rank m is called to have a nonzero signed row compound provided that each square submatrix of order m of A is an SNS-matrix or has an identically zero determinant. As generalizations of SNS-matrices, S⁎-matrices, and totally L-matrices, matrices with nonzero signed row compound have a close relationship with matrices with signed null-spaces which are applied to characterize linear systems with signed solutions. In this paper, matrices with nonzero signed row compound are characterized in terms of their signed bipartite graphs. Following these results, characterizations of matrices with signed null-spaces and full row term ranks in terms of their signed bipartite graphs are obtained too. The recursive structure of m×(m+2) row sign balanced (RSB) maximal (0,1,−1)-matrices with nonzero signed row compound (or with signed null-spaces) is also characterized.