THE LINEAR COMPLEMENTARITY-PROBLEM WITH EXACT ORDER MATRICES

A real n by n matrix A is called an N(P)-matrix of exact order k, if the principal minors of A of order 1 through (n - k) are negative (positive) and (n - k + 1) through n are positive (negative). In this paper the properties of exact order 1 and 2 matrices are investigated, using the linear complementarity problem LCP(q, A) for each q is-an-element-of R(n). A complete characterization of the class of exact order 1 based on the number of solutions to the LCP(q, A) for each q is-an-element-of R(n) is presented. In the last section we consider the problem of computing a solution to the LCP(q, A) when A is a matrix of exact order 1 or 2.