Let \[\mathbf{A}_1=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}, \qquad \mathbf{A}_2=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \qquad \mathbf{A}_3 = \begin{pmatrix}0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{pmatrix}.\] Find in each case an orthogonal matrix \(\mathbf{R}\) such that \(\mathbf{R}\mathbf{A}_i\mathbf{R}^T\) is diagonal.

Let \(S^{n-1}=\{\vec v\in \mathbb R^n | \|\vec v\|=1\},\) where \(\Vert\cdot\Vert\) denotes the norm induced by the standard scalar product on \(\mathbb{R}^n.\) For a symmetric matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) with eigenvalues \(\lambda_1\geqslant \lambda_2 \geqslant \ldots \geqslant \lambda_n\) we consider the function \[q: S^{n-1} \to \mathbb{R}, \qquad \vec v\mapsto \langle \vec v,\vec v\rangle_\mathbf{A}.\] Show that \(\max_{\vec{v} \in S^{n-1}}\{q(\vec v)\} = \lambda_1\) and \(\min_{\vec{v} \in S^{n-1}}\{q(\vec v)\} = \lambda_n.\)

Show that if a matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) is orthogonal, symmetric and positive definite, then \(\mathbf{A}\) must be the identity matrix.

Let \((V,\langle \cdot{,}\cdot\rangle)\) be a finite dimensional Euclidean space and \(g : V \to V\) an endomorphism. Show that \[\operatorname{Ker}g=(\mathrm{Im}g^*)^{\perp}.\]