I wonder if the following type of matrices have been studied before, and if there's an expression for its eigenvalues and eigenvectors.
Fix a positive integer $m$ and let $T_m$ be the $n \times n$ matrix $(t_{j,k})$ given by $t_{j,k} = a_{mk - j}$ for some sequence $(a_j)$ satisfying $a_{j-n} = a_j$ for all $j$. Note when $m = 1$, that $T_1$ is a regular circulant matrix. Have you guys seen these types of matrices before? Could we have an expression for its eigenvalues and eigenvectors, similar to the regular circulants? Thanks.