Let $O$ be an orthogonal matrix of $M _ { n } ( \mathbb { R } )$ and $S$ a sign diagonal matrix. Show that the equality $O x = S x$ with $x \in \mathbb { R } ^ { n }$ strictly positive is equivalent to
$$( * ) \left\{ \begin{array} { c }
\left( I _ { n } + O \right) x \geq 0 \\
\left( I _ { n } - O \right) x \geq 0 \\
x > 0
\end{array} \right.$$