A note on the eigenvalues of double band matrices

TL;DR

This paper analyzes the eigenvalues of matrices with two diagonal bands of positive entries, revealing they are scaled roots of unity determined by the matrix's period, which depends on the band distance.

Contribution

It provides a novel characterization of eigenvalues for double band matrices, linking them to roots of unity and the matrix's period, a new insight in matrix spectral theory.

Findings

Eigenvalues are of the form r * ζ, with ζ a pth root of unity.

Eigenvalues are scaled by a nonnegative real number r.

The period p is computed from the distance between the bands.

Abstract

We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form $r \zeta$, where $r$ is a nonnegative real number and $\zeta$ is a $p$th root of unity, where $p$ is the period of the matrix, which is computed from the distance between the bands.