Consider $a_0, \ldots, a_{k-1} \in \mathbb{R}$, consider matrix $\mathbf{A}$ as the following
$$\mathbf{A} = \begin{bmatrix} a_{0} & a_{1} & \ldots & a_{k-1} \\ a_{1} & a_{0} & \ldots &a_{k-2} \\ \vdots & \vdots & \vdots & \vdots \\ a_{k-1} & a_{k-2} & \ldots& a_{0} \end{bmatrix} = \mathbf{A}^\top$$
Clearly, this matrix is symmetric, or hermitian, so it has a nice diagonal eigenvalue decomposition. Just curious what are the properties about this matrix.