The Eigen-Problem for Some Special Near-Toeplitz Centro-Skew Tridiagonal Matrices

TL;DR

This paper derives the eigenvalues and eigenvectors of a specific class of near-Toeplitz centro-skew tridiagonal matrices, providing explicit solutions for their spectral properties.

Contribution

It presents an explicit eigen-decomposition for a special class of near-Toeplitz centro-skew tridiagonal matrices, which was previously not well-understood.

Findings

Eigenvalues are explicitly characterized.

Eigenvectors are explicitly constructed.

Spectral properties are fully determined.

Abstract

Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the eigen-pairs of the matrices $R_n$.