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 we choose the root $t_0$ of (1). We define $$R = \sum_{i,j=1}^{n} \left|s_{jj} - s_{ii}\right| \quad \text{and} \quad R' = \sum_{i,j=1}^{n} \left|s_{jj}' - s_{ii}'\right|$$ Show that $$R' - R \geqslant 2\left(\left|s_{qq}' - s_{qq}\right| + \left|s_{pp}' - s_{pp}\right|\right) = 2\sum_{i=1}^{n} \left|s_{ii}' - s_{ii}\right|$$
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 we choose the root $t_0$ of (1).
We define
$$R = \sum_{i,j=1}^{n} \left|s_{jj} - s_{ii}\right| \quad \text{and} \quad R' = \sum_{i,j=1}^{n} \left|s_{jj}' - s_{ii}'\right|$$
Show that
$$R' - R \geqslant 2\left(\left|s_{qq}' - s_{qq}\right| + \left|s_{pp}' - s_{pp}\right|\right) = 2\sum_{i=1}^{n} \left|s_{ii}' - s_{ii}\right|$$