1.II.7B

(i) Let $\mathbf{u}, \mathbf{v}$ be unit vectors in $\mathbb{R}^{3}$. Write the transformation on vectors $\mathbf{x} \in \mathbb{R}^{3}$

$\mathbf{x} \mapsto(\mathbf{u} \cdot \mathbf{x}) \mathbf{u}+\mathbf{v} \times \mathbf{x}$

in matrix form as $\mathbf{x} \mapsto A \mathbf{x}$ for a matrix $A$. Find the eigenvalues in the two cases (a) when $\mathbf{u} \cdot \mathbf{v}=0$, and (b) when $\mathbf{u}, \mathbf{v}$ are parallel.

(ii) Let $\mathcal{M}$ be the set of $2 \times 2$ complex hermitian matrices with trace zero. Show that if $A \in \mathcal{M}$ there is a unique vector $\mathrm{x} \in \mathbb{R}^{3}$ such that

$A=\mathcal{R}(\mathbf{x})=\left(\begin{array}{cc} x_{3} & x_{1}-i x_{2} \\ x_{1}+i x_{2} & -x_{3} \end{array}\right)$

Show that if $U$ is a $2 \times 2$ unitary matrix, the transformation

$A \mapsto U^{-1} A U$

maps $\mathcal{M}$ to $\mathcal{M}$, and that if $U^{-1} \mathcal{R}(\mathbf{x}) U=\mathcal{R}(\mathbf{y})$, then $\|\mathbf{x}\|=\|\mathbf{y}\|$ where $\|\cdot\|$ means ordinary Euclidean length. [Hint: Consider determinants.]