On Column Competent Matrices and Linear Complementarity Problem

TL;DR

This paper explores the properties of column competent matrices and their role in ensuring solution uniqueness in linear complementarity problems, providing new theoretical insights and connections to other matrix classes.

Contribution

It introduces new results on the $w$-uniqueness of solutions related to column competent matrices and links them to column adequate matrices using degree theory.

Findings

Characterization of $w$-uniqueness via column competent matrices

New theorems connecting column competent and column adequate matrices

Implications for algorithms in operations research

Abstract

We revisit the class of column competent matrices and study some matrix theoretic properties of this class. The local $w$-uniqueness of the solutions to the linear complementarity problem can be identified by the column competent matrices. We establish some new results on $w$-uniqueness properties in connection with column competent matrices. These results are significant in the context of matrix theory as well as algorithms in operations research. We prove some results in connection with locally $w$-uniqueness property of column competent matrices. Finally we establish a connection between column competent matrices and column adequate matrices with the help of degree theory.