The Linear Complementarity Problem with Exact Order Matrices

(Dedicated to Professor K. G. Ramamurthy on the occasion of his 60th birthday.)

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 ∈ Rⁿ. A complete characterization of the class of exact order 1 based on the number of solutions to the LCP(q, A) for each q ∈ Rⁿ is presented. In the last season we consider the problem of computing a solution to the LCP(q, A) when A is a matrix of exact order 1 or 2.