(a) Show that the eigenvalues of any real $n \times n$ square matrix $A$ are the same as the eigenvalues of $A^{T}$.

The eigenvalues of $A$ are $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ and the eigenvalues of $A^{T} A$ are $\mu_{1}, \mu_{2}, \ldots$, $\mu_{n}$. Determine, by means of a proof or a counterexample, whether the following are necessary valid: (i) $\sum_{i=1}^{n} \mu_{i}=\sum_{i=1}^{n} \lambda_{i}^{2}$; (ii) $\prod_{i=1}^{n} \mu_{i}=\prod_{i=1}^{n} \lambda_{i}^{2}$.

(b) The $3 \times 3$ matrix $B$ is given by

$$B=I+\mathbf{m n}^{T}$$

where $\mathbf{m}$ and $\mathbf{n}$ are orthogonal real unit vectors and $I$ is the $3 \times 3$ identity matrix.

(i) Show that $\mathbf{m} \times \mathbf{n}$ is an eigenvector of $B$, and write down a linearly independent eigenvector. Find the eigenvalues of $B$ and determine whether $B$ is diagonalisable.

(ii) Find the eigenvectors and eigenvalues of $B^{T} B$.