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

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 $x\mapsto (1\mathrm{/}2)(x+E_{n}x)$ (respectively, $x\mapsto (1\mathrm{/}2)(x-E_{n}x)$). Then $R_n = (\pi_n - \pi_n^T) E_+ + (\eta_n - \eta_n^T) E_- ,$where$\pi_n$isthebasic$n\times n$circulantmatrixand$\eta_n$isthebasic$n\times n$skew-circulantmatrix\mathrm{.}Inotherwords,if$x$isavectorintherangeof$E_+$then$R_n x = (\pi_n - \pi_n^T)x$andif$x$isintherangeof$E_-$then$R_n x = (\eta_n - \eta_n^T)x$.