Prove that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal (i.e. $\mathbf{e}_{i}^{*} \cdot \mathbf{e}_{j}=0$ ).

Let $A$ be a real $3 \times 3$ non-zero antisymmetric matrix. Show that $i A$ is Hermitian. Hence show that there exists a (complex) eigenvector $\mathbf{e}_{1}$ such $A \mathbf{e}_{1}=\lambda \mathbf{e}_{1}$, where $\lambda$ is imaginary.

Show further that there exist real vectors $\mathbf{u}$ and $\mathbf{v}$ and a real number $\theta$ such that

$$A \mathbf{u}=\theta \mathbf{v} \quad \text { and } \quad A \mathbf{v}=-\theta \mathbf{u}$$

Show also that $A$ has a real eigenvector $\mathbf{e}_{3}$ such that $A \mathbf{e}_{3}=0$.

Let $R=I+\sum_{n=1}^{\infty} \frac{A^{n}}{n !}$. By considering the action of $R$ on $\mathbf{u}, \mathbf{v}$ and $\mathbf{e}_{3}$, show that $R$ is a rotation matrix.