On Bounds for the Smallest and the Largest Eigenvalues of GCD and LCM Matrices

TL;DR

This paper derives bounds for the smallest and largest eigenvalues of GCD and LCM matrices defined on the set {1, 2, ..., n}, using specific arithmetical functions.

Contribution

It provides new upper and lower bounds for eigenvalues of GCD and LCM matrices based on arithmetical functions, enhancing understanding of their spectral properties.

Findings

Established bounds for eigenvalues of GCD matrices

Established bounds for eigenvalues of LCM matrices

Bound expressions involve arithmetical functions

Abstract

In this paper, we study the eigenvalues of the GCD matrix $(S_n)$ and the LCM matrix $[S_n]$ defined on $S_n=\{1,2,\ldots,n\}$. We present upper and lower bounds for the smallest and the largest eigenvalues of $(S_n)$ and $[S_n]$ in terms of particular arithmetical functions.