ON STEIN TRANSFORMATION IN SEMIDEFINITE LINEAR COMPLEMENTARITY PROBLEMS

Abstract

In the setting of semidenite linear complementarity problems on Sn, we focus on the Stein Transformation SA(X) := X −AXAT , and show that SA is (strictly) monotone if and only if r(UAUT ◦UAUT ) (< ) ≤ 1 for all orthogonal matrices U where ◦is the Hadamard product and r is the real numerical radius. In particular, we show that if (A) < 1 and r(UAUT ◦UAUT ) ≤ 1, then SDLCP(SAQ) has a unique solution for all Q ∈ Sn. In an attempt to characterize the GUS-property of a nonmonotone SA, we give an instance of a nonnormal 2 × 2 matrix A such that SDLCP(SAQ) has a unique solution for Q either a diagonal or a symmetric positive or negative semidenite matrix. We show that this particular SA has the P′ 2-property.