On the positive definiteness and eigenvalues of meet and join matrices

TL;DR

This paper investigates the positive definiteness and eigenvalues of meet and join matrices, providing necessary and sufficient conditions, graph-theoretic insights, and bounds for eigenvalues based on function properties.

Contribution

It introduces a novel approach to characterize positive definiteness of meet and join matrices, including graph-theoretic conditions and eigenvalue bounds, extending existing theory.

Findings

Necessary and sufficient conditions for positive definiteness when set is meet closed

Graph-theoretic assumptions reduce conditions on the function f

Bounds for eigenvalues based on monotonicity of f

Abstract

In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set $S_n$ is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix $(S_n)_f$. From this condition we obtain some sufficient conditions for positive definiteness as corollaries. We also use graph theory and show that by making some graph theoretic assumptions on the set $S_n$ we are able to reduce the assumptions on the function $f$ while still preserving the positive definiteness of the matrix $(S_n)_f$. Dual theorems of these results for join matrices are also presented. As examples we consider the so-called power GCD and power LCM matrices as well as MIN and MAX matrices. Finally we give bounds for the eigenvalues of meet and join matrices in cases when the function $f$ possesses certain monotonic behaviour.