Let $A \in \mathcal{M}_n(\mathbb{R})$. We propose to give a necessary and sufficient condition for the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ to the system
$$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
Show that if the system $(*)$ admits a solution in $\mathrm{GL}_n(\mathbb{R})$, then the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.