On semidefinite linear complementarity problems

Abstract

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∈R n×n , we consider the linear transformation L A :? n →? n defined by L A (X):=AX+XA T and show that the P- and Q-properties for L A 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.