We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal and $a_i \in \mathbb{Q}$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$, and $P \in \mathrm{GL}_d(\mathbb{Q})$, $q_1, \ldots, q_d \in \mathbb{Q}$, $q_i > 0$ such that $S = P^T \cdot \operatorname{Diag}(q_1, \ldots, q_d) \cdot P$. We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$
17a. Verify that the matrix $S M$ is symmetric.
17b. Deduce that the matrix $R M R ^ { - 1 }$ is symmetric where $R = \operatorname { Diag } \left( \sqrt { q _ { 1 } } , \ldots , \sqrt { q _ { d } } \right) \cdot P$.