Generalized GCD matrices

TL;DR

This paper introduces a generalized GCD matrix where each element is defined by an arithmetical function evaluated at the index and their gcd, expanding the classical GCD matrix concept.

Contribution

It extends the classical GCD matrix by considering elements as functions of the index and their gcd, providing a broader framework for analysis.

Findings

Provides properties of the generalized GCD matrix

Establishes conditions for invertibility and eigenvalues

Connects to classical GCD matrix results

Abstract

Let $f$ be an arithmetical function. The matrix $[f(i,j)]_{n\times n}$ given by the value of $f$ in greatest common divisor of $(i,j)$, $f\big((i,j)\big)$ as its $i,\; j$ entry is called the greatest common divisor (GCD) matrix. We consider the generalization of this matrix where the elements are in the form $f\big(i,(i,j)\big)$.