Isotone projection cones and Q-matrices

TL;DR

This paper explores the use of isotone projection cones to generate a broad class of Q-matrices, which are crucial for solving complementarity problems, by establishing a link between certain matrix products and Q-matrix properties.

Contribution

It introduces a novel method of constructing Q-matrices using isotone projection cones, specifically through the product of non-negative matrices with positive diagonals and Stieltjes matrices.

Findings

Product of a non-negative matrix with positive diagonal and a Stieltjes matrix is a Q-matrix.

Isotone projection cones can be used to generate large classes of Q-matrices.

The approach aids in solving complementarity problems and variational inequalities.

Abstract

Proper cones with the property that the projection onto them is isotone with respect to the order they induce are called isotone projection cones. Isotone projection cones and their extensions have been used to solve complementarity problems and variational inequalities. Q-matrices are matrices with the property that all classical linear complementarity problems defined by them are solvable. This note will use the isotone projection cones to generate a large class of Q-matrices. More specifically, it will be shown that the product between a non-negative matrix with positive diagonal elements and a Stieltjes matrix is a Q-matrix.