We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d})$ with $\alpha_{i} \geqslant 0$ in decreasing order, $U = (U_{ij})_{1 \leqslant i,j \leqslant d}$, and $S_{dd}$ as defined in question 24.
[(a)] Show that $S_{dd} = \sum_{j=1}^{d} \alpha_{j} U_{jd}^{2}$ where $U = (U_{ij})_{1 \leqslant i,j \leqslant d}$.
[(b)] Deduce that $\langle D, R \rangle \leqslant \left(\sum_{i=1}^{d-1} \alpha_{i}\right) - \alpha_{d}$.
We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d})$ with $\alpha_{i} \geqslant 0$ in decreasing order, $U = (U_{ij})_{1 \leqslant i,j \leqslant d}$, and $S_{dd}$ as defined in question 24.
\begin{itemize}
\item[(a)] Show that $S_{dd} = \sum_{j=1}^{d} \alpha_{j} U_{jd}^{2}$ where $U = (U_{ij})_{1 \leqslant i,j \leqslant d}$.
\item[(b)] Deduce that $\langle D, R \rangle \leqslant \left(\sum_{i=1}^{d-1} \alpha_{i}\right) - \alpha_{d}$.
\end{itemize}