The linear complementarity problem and a subclass of fully semimonotone matrices (Q1087473)

We investigate a subclass \({\mathcal W}\) of the \(n\times n\) real matrices. A matrix M belongs to \({\mathcal W}\) if and only if certain pairs of its complementary cones intersect only in the zero vector. We show that \({\mathcal W}\subseteq {\mathcal U}\), where \({\mathcal U}\) is the class of matrices M for which LCP(q,M) (the linear complementarity problem) has a unique solution whenever q belongs in the interior of the union of its complementary cones. Membership in \({\mathcal W}\) can be established by solving \(2^{n-1}\) systems of linear equations. Finally, several sufficient conditions are given on a matrix in \({\mathcal W}\) so that it possesses nonnegative principal minors.