Necessary and sufficient conditions for the existence of complementary solutions and characterizations of the matrix classes \(Q\) and \(Q_ 0\) (Q1177232)

Let \(Q\) \((Q_ 0)\) be the set of all real matrices \(M\) such that the linear complementarity problem (denoted by \((q,M))\) \(Mx+q\geq 0\), \(x\geq 0\), \(x^ T(Mx+q)=0\) has a solution \(x\) for any real (feasible) \(q\). For \(M\) given, \(q\) is called feasible whenever there is an \(x\geq 0\) such that \(Mx+q\geq 0\). Starting from a result by Mangasarian the author gives a new characterization of \(Q\) and \(Q_ 0\). He also shows that for positive definite (semidefinite) matrices \(M\in Q\) \((M\in Q_ 0)\) can be checked by a finite set of computable conditions.