On some properties of \(\mathbf P\)-matrix sets
Let \({\mathcal C}\) be a non-empty set of matrices in \(\mathbb{R}^{n\times n}\). The paper introduces the concept of row-P-property for the set of matrices in \({\mathcal C}\). It is shown that this property is equivalent to the fact that for any \(x\neq 0\) in \(\mathbb{R}^n\), there exists an index \(i\) with \(x_i(M x)_i > 0\) for all \(M\in{\mathcal C}\). It is also shown that a compact convex set \({\mathcal C}\) in \(\mathbb{R}^{n\times n}\) has the row-P-property if and only if the set of extreme point of \({\mathcal C}\) has the row-P-property. It is also proved that a compact set \({\mathcal C}\) in \(\mathbb{R}^{n\times n}\) has the row-P-property if and only if the complementarity problem \(CP({\mathcal C},q)\) has a unique solution for every compact \(q\) in \(\mathbb{R}^n\). As a corollary, the existence of a unique \(u>0\) such that \(Mu>0\) for all \(M\in{\mathcal C}\) is obtained which is shown to fail without the row-P-property.
scientific article; zbMATH DE number 1331376