I was reading a proof utilizing some property of the spectral norm, but fail to understand some steps. The part of the proof goes like,
\begin{equation*} \begin{split} \left\|\left[\begin{array}{cc}{\sin ^{2} \Theta} & {-\cos \Theta \sin \Theta} \\ {-\cos \Theta \sin \Theta} & {-\sin ^{2} \Theta}\end{array}\right]\right\| =& \max _{1 \leq i \leq r}\left\|\left[\begin{array}{cc}{\sin ^{2} \theta_{i}} & {-\cos \theta_{i} \sin \theta_{i}} \\ {-\cos \theta_{i} \sin \theta_{i}} & {-\sin ^{2} \theta_{i}}\end{array}\right]\right\|\\[0.5cm] = &\max _{1 \leq i \leq r}\left\|\sin \theta_{i}\left[\begin{array}{cc}{\sin \theta_{i}} & {-\cos \theta_{i}} \\ {-\cos \theta_{i}} & {-\sin \theta_{i}}\end{array}\right]\right\|\\[0.5cm] =& \max _{1 \leq i \leq r}\left|\sin \theta_{i}\right| \end{split} \end{equation*} where $|| \cdot||$ means the spectral norm.
$\sin ^{2}\Theta$ and $-\cos \Theta \sin \Theta$ are diagonal matrix with $\sin^2\theta_i$ and $-\cos\theta_i\sin\theta_i$ as its diagonal elements.
Can anyone help me to explain these few steps?
Note that there exists a permutation matrix $P$ such that $$ P\pmatrix{ {\sin ^{2} \Theta} & {-\cos \Theta \sin \Theta} \\ {-\cos \Theta \sin \Theta} & {-\sin ^{2} \Theta} }P^T = \pmatrix{M_1 \\ & \ddots \\ && M_r} $$ where $$ M_i = \pmatrix{\sin^2 \theta_i & - \sin \theta_i \cos \theta_i \\ - \sin \theta_i \cos \theta_i & - \sin \theta_i}. $$ From there, we have $$ \left\|\pmatrix{ {\sin ^{2} \Theta} & {-\cos \Theta \sin \Theta} \\ {-\cos \Theta \sin \Theta} & {-\sin ^{2} \Theta} }\right\| = \\ \left\|P\pmatrix{ {\sin ^{2} \Theta} & {-\cos \Theta \sin \Theta} \\ {-\cos \Theta \sin \Theta} & {-\sin ^{2} \Theta} }P^T\right\| = \\ \left\|\pmatrix{M_1 \\ & \ddots \\ && M_r}\right\| = \max_{1 \leq i \leq r} \|M_i\|. $$