On Positive Semidefinite Preserving Stein Transformation

Abstract

In the setting of semidefinite linear complementarity problems on $\sn$,we focus on the Stein Transformation $S_A(X):= X - AXA^T$ for $A\in \Rnn$ that is positive semidefinite preserving (i.e., $S_A(\sn_+) \subseteq \sn_+$) and show that such transformation is strictly monotone if and only if it is nondegenerate. We also show that a positive semidefinite preserving $S_A$ has the Ultra-GUS property if and only if $1 \not\in \sigma(A)\sigma(A).$