A Complementarity Property of Rank-One Multiplicative Maps

The semidefinite linear complementarity problem associated with the multiplicative map $M_A:\mathbb{S}^n\to\mathbb{S}^n$ defined by $M_A(X)=AXA^T$, is considered, in the case when $A\in\mathbb{R}^{n\times n}$ has rank one. A linear map is said to have the $Q_0$-property if every feasible instance of