Kac Matrix

The $(n+1)\times(n+1)$ Tridiagonal Matrix (also called the Clement Matrix) defined by

$$S_n=\left[{\matrix{ 0 & n & 0 & 0 & \cdots & 0\cr 1 & 0 & ... ...0 & 0 & 0 & n-1 & 0 & 1\cr 0 & 0 & 0 & 0 & n & 0\cr}}\right].$$

The Eigenvalues are $2k-n$ for $k=0$, 1, ..., $n$.