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

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 $R_{n}x=({\eta }_n-{\eta }_n^T)x$.