Some new results for the semidefinite linear complementarity problem

In this paper, we present some new results for the semidefinite linear complementarity problem ( SDLCP). In the first part, we introduce the concepts of ( i) nondegeneracy for a linear transformation L : S-n -->S-n and (ii) the locally- star- like property of a solution point of an SDLCP( L, Q) for Q is an element of S-n, and we relate them to the finiteness of the solution set of SDLCP( L, Q) as Q varies in S-n. In the second part, we show that for positive stable matrices A (1),..., A(k), the linear transformation L : = L (A1) o L (A2) o...o L (Ak) has the Q - property where L (Ai) (X) : = A(i) X + X A(i)(T). A similar result is proved for the transformation S : = S (A1) o S (A2) o...o S (Ak), where each A(i) is Schur stable and S (Ai) (X) : = X - A(i)XA(i)(T). We relate these results to the simultaneous stability of a finite set of matrices.