We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $R(E_{1}) = E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$. We consider a matrix $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d}) \in \mathscr{M}_{d}(\mathbb{R})$ diagonal with diagonal entries $\alpha_{i} \geqslant 0$ in decreasing order. We denote $U = (u_{1} | \ldots | u_{d})$.
[(a)] Verify that $\langle D, R \rangle = \langle S, R^{\prime} \rangle$ where $R^{\prime} = U^{T}RU$ and $S = U^{T}DU$.
[(b)] Show that if $R_{0} = (R_{ij}^{\prime})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$ then $R_{0} \in \mathrm{O}_{d-1}(\mathbb{R})$.
We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $R(E_{1}) = E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$. We consider a matrix $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d}) \in \mathscr{M}_{d}(\mathbb{R})$ diagonal with diagonal entries $\alpha_{i} \geqslant 0$ in decreasing order. We denote $U = (u_{1} | \ldots | u_{d})$.
\begin{itemize}
\item[(a)] Verify that $\langle D, R \rangle = \langle S, R^{\prime} \rangle$ where $R^{\prime} = U^{T}RU$ and $S = U^{T}DU$.
\item[(b)] Show that if $R_{0} = (R_{ij}^{\prime})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$ then $R_{0} \in \mathrm{O}_{d-1}(\mathbb{R})$.
\end{itemize}