Summary: 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 \(L:{\mathcal S}^n \rightarrow{\mathcal S}^n\) and (ii) the locally-star-like property of a solution point of an SDLCP(\(L,Q\)) for \(Q\in{\mathcal S}^n\), and we relate them to the finiteness of the solution set of SDLCP(\(L,Q\)) as \(Q\) varies in \({\mathcal S}^n\). In the second part, we show that for positive stable matrices \(A_1,\dots , A_k\), the linear transformation \(L:=L_{A_1} \circ L_{A_2} \circ \cdots \circ L_{A_k} \) has the Q-property where \(L_{A_i}(X):= A_i X + XA_i^T\). A similar result is proved for the transformation \(S:=S_{A_1} \circ S_{A_2} \circ \cdots \circ S_{A_k}\), where each \(A_i\) is Schur stable and \(S_{A_i}(X):=X-A_i XA_i^T\). We relate these results to the simultaneous stability of a finite set of matrices.