A Complementarity Property of Rank-One Multiplicative Maps

Abstract

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 the associated complementarity problem admits a complementary solution. We prove two sharp, complementary results. When $A=xy^T$ with $x,y\in\mathbb{R}^n$ \emph{linearly independent}, we construct a matrix $Q\in\mathbb{S}^n$ for which the associated semidefinite complementarity problem is feasible, but possesses no complementary solution; proving that $M_A$ \emph{does not have} the $Q_0$-property. Conversely, when $A=uu^T$ for some nonzero $u\in\mathbb{R}^n$, we prove that $M_A$ \emph{has} the $Q_0$-property. These two results provide a complete characterization of the $Q_0$-property within the family of rank-one multiplicative maps.