Matrices of Sign-Solvable Linear Systems

S*-matrices and SNS*-matrices

We first recall some basic facts from sections 1.2 and 2.1. Let A be an m by m + 1 matrix. By definition, A is an S*-matrix if and only if each submatrix of A of order m is an SNS-matrix. By (iv) of Theorem 2.1.1, A is an S*-matrix if and only if there exists A strict signing D such that AD and A(–D) are the only column signings of A each of whose rows are balanced. The matrix A is an S-matrix if and only if A is an S*-matrix and AIm+1 = A and A(–Iw+1) = –A are the only column signings of A each of whose rows are balanced.

Let v1, v2, … vm+1 be the column vectors of A. Then v1, v2,…, vm+1 are the vertices of an m -simplex whose interior contains the origin if and only if the right null space of A is spanned by A vector w each of whose entries is positive. This implies A geometric description of S-matrices [6]:

Clearly, each row of an S*-matrix must contain at least two nonzero entries. The following theorem [7] shows that an m by m + 1 matrix with at least three nonzero entries in each row is not an S*-matrix.