Let banded Toeplitz matrix $W\in \mathbb{R}^{n\times n}$ be defined by $$W_{jk} = \begin{cases} m - |j-k| & \text{ if } |j-k| \leq m \\ 0 & \text{ if } |j-k| > m \end{cases}$$ Can one get a closed-form eigenvalue of $W$?
I find that the matrix $W$ can be written as $XX^T$ where $X$ is the following $n \times (n+m-1)$ Toeplitz matrix
\begin{gather*} X = \begin{pmatrix} 1 & 1 & 1 & \dots & 1 & 0 & \dots & 0\\ 0 & 1 & 1 & 1 & \dots & 1 & 0\\ 0 & 0 & 1 & 1 & 1 & \dots & 1 & \dots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 0 & 1 & \dots &1 \end{pmatrix} \end{gather*}
but I cannot find any theory dealing with the singular value of a matrix like $X$.