Ky Fan's N-matrices and linear complementarity problems

We consider the linear complementarity problem (LCP),w=Az + q, w0,z0,w^Tz=0, when all the off-diagonal entries ofA are nonpositive (the class of Z-matrices), all the proper principal minors ofA are positive and the determinant ofA is negative (the class of almost P-matrices). We shall call this the class of F-matrices. We show that ifA is a Z-matrix, thenA is an F-matrix if and only if LCP(q, A) has exactly two solutions for anyq0,q0, and has at most two solutions for any otherq.Research supported by AFOSR-89-0512.