Linear complementarity properties of some classes of banded matrices

TL;DR

This paper investigates the linear complementarity properties of various banded matrices, including triangular and bidiagonal southwest matrices, and extends the analysis to matrix transformations in Euclidean Jordan algebras.

Contribution

It characterizes the Q-property for specific classes of banded matrices and extends the analysis to matrix transformations in Euclidean Jordan algebras, including all 2x2 Q-matrices.

Findings

Characterization of Q-property for triangular and bidiagonal southwest matrices

Complete description of 2x2 Q-matrices

Conditions for rank-one transformations to have the Q-property

Abstract

A banded matrix is a real square matrix where nonzero entries appear around the main diagonal. In this article, we consider linear complementarity properties of (variants) of banded matrices. Focusing on triangular matrices and the newly defined bidiagonal southwest matrices, we describe several results characterizing the Q-property in terms of the sign patterns and determinant of the given matrix. As a byproduct, we describe all Q-matrices of size 2 by 2. Extending these results to Euclidean Jordan algebras, we consider matrix-based linear transformations and study the Q-property. In particular, we show that a rank-one linear transformation of the form a\otimes b has the Q-property if and only if either a>0,b>0, or a<0, b<0.