Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r)

TL;DR

This paper systematically analyzes the asymptotic behavior of the smallest eigenvalue of matrices linked to completely even functions modulo r, including Smith matrices, providing new theorems and illustrative examples.

Contribution

It introduces new theorems on the asymptotic behavior of eigenvalues for matrices associated with completely even functions mod r, including Smith matrices.

Findings

Asymptotic formulas for the smallest eigenvalue derived

Theorems established for matrices related to completely even functions

Examples demonstrate the main theoretical results

Abstract

In this paper we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod $r$). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod $r$). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results.