Let $A_{1}, A_{2}, \ldots, A_{k}$ be $n \times n$ matrices over a field $\mathbb{F}$. We say $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable if there exists an invertible matrix $P$ such that $P^{-1} A_{i} P$ is diagonal for all $1 \leqslant i \leqslant k$. We say the matrices are commuting if $A_{i} A_{j}=A_{j} A_{i}$ for all $i, j$.

(i) Suppose $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable. Prove that they are commuting.

(ii) Define an eigenspace of a matrix. Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting $n \times n$ matrices over a field $\mathbb{F}$. Let $E$ denote an eigenspace of $B_{1}$. Prove that $B_{i}(E) \leqslant E$ for all $i$.

(iii) Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting diagonalisable matrices. Prove that they are simultaneously diagonalisable.

(iv) Are the $2 \times 2$ diagonalisable matrices over $\mathbb{C}$ simultaneously diagonalisable? Explain your answer.