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 $E_n(i,j):=δ(i+j,n+1)$. 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 $R_n = (π_n - π_n^T) E_+ + (η_n - η_n^T) E_- ,$ where $π_n$ is the basic $n\times n$ circulant matrix and $η_n$ is the basic $n\times n$ skew-circulant matrix. In other words, if $x$ is a vector in the range of $E_+$ then $R_n x = (π_n - π_n^T)x$ and if $x$ is in the range of $E_-$ then $R_n x = (η_n - η_n^T)x$.

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