In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set
$$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$
Let $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. We denote by $Q \in \mathbb{R}[X]$ the polynomial $\sum_{k=0}^{n-1} a_{k} X^{k}$.
What are the complex eigenvalues of the matrix $J(a_{0}, \ldots, a_{n-1})$?