Karamardian Matrices: A Generalization of Q-Matrices
Abstract
A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a solution for all q ∈ Rn. This means that for every vector q there exists a vector x such that x ≥ 0, y=Ax+q≥ 0 and xTy=0. A well known result of Karamardian states that if the problems LCP(A,0) and LCP(A,d) for some d∈ Rn, d >0 have only the zero solution, then A is a Q-matrix. By relaxing the condition on d and imposing a condition on the solution vector x in the two problems as above, the authors introduce a new class of matrices called Karamardian matrices, requiring that these two modified problems have only zero as a solution. In this article, a systematic treatment of Karamardian matrices is undertaken. Among other things, it is shown how Karamardian matrices have properties that are analogous to those of Q-matrices. A subclass of a recently introduced notion of P\#-matrices is shown to possess the Karamardian property, and for this reason we undertake a thorough study of P\#-matrices and make some fundamental contributions.