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A class of singular R0 matrices and extensions to semidefinite linear complementarity problems

In this article, the class of R0-matrices is extended to include typically singular matrices, by requiring in addition that the solution x above belongs to a subspace of Rn. This idea is then extended to semidefinite linear complementarity problems, where a characterization is presented for the multplicative transformation. | Yugoslav Journal on Operations Research 23(2013) Number 2, 163–172 DOI: 10.2298/YJOR130105027S A CLASS OF SINGULAR R0 -MATRICES AND EXTENSIONS TO SEMIDEFINITE LINEAR COMPLEMENTARITY PROBLEMS Koratti Chengalrayan SIVAKUMAR Department of Mathematics Indian Institute of Technology Madras Chennai 600 036, India. Received: January, 2013 / Accepted: March, 2013 Abstract: For A ∈ Rn×n and q ∈ Rn , the linear complementarity problem LCP (A, q) is to determine if there is x ∈ Rn such that x ≥ 0, y = Ax + q ≥ 0 and xT y = 0. Such an x is called a solution of LCP (A, q). A is called an R0 -matrix if LCP (A, 0) has zero as the only solution. In this article, the class of R0 -matrices is extended to include typically singular matrices, by requiring in addition that the solution x above belongs to a subspace of Rn . This idea is then extended to semidefinite linear complementarity problems, where a characterization is presented for the multplicative transformation. Keywords: R0 -matrix; semidefinite linear complementarity problems; MoorePenrose inverse; group inverse. MSC: 90C33; 15A09 1. INTRODUCTION Let A ∈ Rn×n and q ∈ Rn . The linear complementarity problem LCP (A, q) is to determine if there exists x ∈ Rn such that x ≥ 0, y = Ax + q ≥ 0 and hx, yi = 0, where for u, v ∈ Rn , we have hu, vi = uT v. Throughout this section, 163 164 K. C. Sivakumar / A Class Of Singular R0 -Matrices And Extensions for u ∈ Rn , the notation u ≥ 0 signifies that all the coordinates of u are nonnegative. If B ∈ Rm×n , then B ≥ 0 denotes that all the entries of B are nonnegative. Motivated by questions concerning the existence and uniqueness of LCP (A, q), many matrix classes have been considered in the literature. Let us recall two such classes. A matrix A ∈ Rn×n is called a Q-matrix if LCP (A, q) has a solution for all q ∈ Rn . A ∈ Rn×n is called an R0 -matrix if LCP (A, 0) has zero as the only solution. Just to recall one of the well known results for an R0 -matrix, we point out that A is