A relation between some special centro-skew, near-Toeplitz, tridiagonal matrices and circulant matrices

TL;DR

This paper explores the relationship between a specific class of centro-skew, near-Toeplitz, tridiagonal matrices and circulant matrices, revealing their structural connections through projections and matrix decompositions.

Contribution

It establishes a novel link between certain tridiagonal matrices and circulant/skew-circulant matrices using projection operators and matrix decompositions.

Findings

$R_n$ is expressed in terms of circulant and skew-circulant matrices.

The matrix $R_n$ acts as a difference of circulant matrices on the range of $E_+$.

The matrix $R_n$ acts as a difference of skew-circulant matrices on the range of $E_-$.

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. This paper shows that $R_n$ is closely related to a certain circulant matrix and a certain skew-circulant matrix. More precisely, let $E_n$ denote the exchange matrix which is defined by $E_n(i,j):=\delta(i+j,n+1)$. Let $E_+$ (respectively, $E_-$) be the projection defined by $x\mapsto (1/2)(x + E_n x)$ (respectively, $x\mapsto (1/2)(x - E_n x)$). Then $R_n = (\pi_n - \pi_n^T) E_+ + (\eta_n - \eta_n^T) E_- ,$ where $\pi_n$ is the basic $n\times n$ circulant matrix and $\eta_n$ is the basic $n\times n$ skew-circulant matrix. In other words, if $x$ is a vector in the range of $E_+$ then $R_n x = (\pi_n - \pi_n^T)x$ and if $x$ is in the range of $E_-$ then $R_n x = (\eta_n - \eta_n^T)x$.