We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$. From now on, and until the end of the problem, we choose the root $t_0$ of (1) and thus the angle $\theta_0$, mentioned in Question 8b. (a) Show that $s_{pp} - s_{pp}'$ and $s_{qq}' - s_{qq}$ have the same sign as $s_{qq} - s_{pp}$. (b) If $1 \leqslant i \leqslant n$, show that $$\left|s_{ii} - s_{qq}'\right| + \left|s_{ii} - s_{pp}'\right| - \left|s_{ii} - s_{pp}\right| - \left|s_{ii} - s_{qq}\right| \geqslant 0$$
We are given a matrix $S \in \mathbf{S}_n$, an angle $\theta \in \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$ and integers $1 \leqslant p < q \leqslant n$ such that $s_{pq} \neq 0$. We define $S' = {}^t R_{p,q}(\theta) S R_{p,q}(\theta)$ and denote its coefficients by $s_{ij}'$.
From now on, and until the end of the problem, we choose the root $t_0$ of (1) and thus the angle $\theta_0$, mentioned in Question 8b.
(a) Show that $s_{pp} - s_{pp}'$ and $s_{qq}' - s_{qq}$ have the same sign as $s_{qq} - s_{pp}$.
(b) If $1 \leqslant i \leqslant n$, show that
$$\left|s_{ii} - s_{qq}'\right| + \left|s_{ii} - s_{pp}'\right| - \left|s_{ii} - s_{pp}\right| - \left|s_{ii} - s_{qq}\right| \geqslant 0$$