Let $A \in \mathcal{M}_n(\mathbb{R})$. Consider the system $$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$ We assume in this question that the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$. a) Justify that we can seek the solutions $X$ of $(*)$ in the form $X = UH$, with $U \in \mathrm{O}(n)$ and $H \in \mathcal{S}_n^{++}(\mathbb{R})$. b) Determine $H$. c) Show the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ of $(*)$ belonging to $\mathrm{GL}_n(\mathbb{R})$.
Let $A \in \mathcal{M}_n(\mathbb{R})$. Consider the system
$$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
We assume in this question that the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.
a) Justify that we can seek the solutions $X$ of $(*)$ in the form $X = UH$, with $U \in \mathrm{O}(n)$ and $H \in \mathcal{S}_n^{++}(\mathbb{R})$.
b) Determine $H$.
c) Show the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ of $(*)$ belonging to $\mathrm{GL}_n(\mathbb{R})$.