Ky Fan’s N-matrices and linear complementarity problems

The paper considers the linear complementarity problem (LCP), w=Az+q, w≥0, z≥0, w’Tz=0, when all the off-diagonal entries of A are nonpositive (the class of Z-matrices), all the proper principal minors of A are positive and the determinant of A is negative (the class of almost P-matrices). This will be called the class of F-matrices. It is shown that if A is a Z-matrix, then A is an F-matrix if and only if LCP(q,A) has exactly two solutions for any q≥0, qℝ0, and has at most two solutions for any other q.