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

In the setting of semidefinite linear complementarity problems on , the implications strict monotonicity P2 GUS P are known. Here, P and P2 properties for a linear transformation are respectively defined by: 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 has the P2-property if and only if for every n×n real invertible matrix Q, every principal subtransformation of has the P-property where Based on this, we show that P2, GUS, and P properties coincide for the two-sided multiplication transfor- mation.