In the recent paper on semidefinite linear complementarity problems, Gowda and Song introduced and studied the P-property, P2-property, GUS-property, and strong monotonicity property for linear transformation L: Sn → Sn, where Sn is the space of all symmetric and real n × n matrices. In an attempt to characterize the P2-property, they raised the following two questions: (i) Does the strong monotonicity imply the P2-property? (ii) Does the GUS-property imply the P2-property? In this paper, we show that the strong monotonicity property implies the P2-property for any linear transformation and describe an equivalence between these two properties for Lyapunov and other transformations. We show by means of an example that the GUS-property need not imply the P2-property, even for Lyapunov transformations.
