Inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs

TL;DR

This paper investigates the inertia, positive definiteness, and b5_pb5 norm of GCD and LCM matrices, along with their unitary analogs, using matrix factorizations and arithmetical function convolutions.

Contribution

It provides new insights into the spectral properties and norms of GCD and LCM matrices and introduces their unitary analogs with detailed analysis.

Findings

Characterization of inertia and positive definiteness conditions

Explicit formulas for b5_pb5 norms of the matrices

Analysis of unitary analogs of GCD and LCM matrices

Abstract

Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.