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 $s_{ij}'$. Suppose in this question that $s_{pq}$ is the coefficient of largest absolute value in $E$. (a) Show that $\|E'\| \leqslant \rho \|E\|$ where $\rho < 1$ is a constant that we will make explicit. (b) If we choose the root $t_0$, show furthermore that $\|D' - D\| \leqslant \|E\|$.
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 $s_{ij}'$.
Suppose in this question that $s_{pq}$ is the coefficient of largest absolute value in $E$.
(a) Show that $\|E'\| \leqslant \rho \|E\|$ where $\rho < 1$ is a constant that we will make explicit.
(b) If we choose the root $t_0$, show furthermore that $\|D' - D\| \leqslant \|E\|$.