On some interconnections between strict monotonicity, globally uniquely solvable, and P properties in semidefinite linear complementarity problems

Abstract

In the setting of semidefinite linear complementarity problems on S n , the implications strict monotonicity⇒ P 2⇒ GUS ⇒ P are known. Here, P and P 2 properties for a linear transformation L: S n→ S n are respectively defined by: X∈ S n, XL( X)= L( X) X⪯0⇒ X=0 and X⪰0, Y⪰0, ( X− Y)[ L( X)− L( Y)]( X+ Y)⪯0⇒ X= Y; GUS refers to the global unique solvability in semidefinite linear complementarity problems correspond- ing to L. In this article, we show that the reverse implications hold for any self-adjoint linear transformation, and for normal Lyapunov and Stein transformations. By introducing the concept of a principal subtransformation of a linear transformation, we show that L: S n→ S n has the P 2 -property if and only if for every n× n real invertible matrix Q, every principal subtransformation of L has the P -property where L(X):=Q T L(QXQ T )Q. Based on this, we show that P 2, GUS, and P properties coincide for the two-sided multiplication transfor- mation.