On semidefinite linear complementarity problems

Given a linear transformation L: Sⁿ → Sⁿ and a matrix Q ∈ Sⁿ, where Sⁿ is the space of all symmetric real n × n matrices, we consider the semidefinite linear complementarity problem SDLCP(L, Sⁿ+, Q) over the cone Sⁿ+ of symmetric n × n positive semidefinite matrics. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A ∈ R^{n × n}, we consider the linear transformation LA : Sⁿ → Sⁿ defined by LA (X) := AX + XA^T and show that the P- and Q-properties for LA are equivalent to A are positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.