Given a linear transformation L:?n→?n and a matrix Q∈?n, where ?n is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?n+,Q) over the cone ?n+ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈Rn×n, we consider the linear transformation LA:?n→?n defined by LA(X):=AX+XAT and show that the P- and Q-properties for LA are equivalent to A being 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.
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