Let $\operatorname{Mat}_{n}(\mathbb{C})$ be the vector space of $n$ by $n$ complex matrices.

Given $A \in \operatorname{Mat}_{n}(\mathbb{C})$, define the linear $\operatorname{map}_{A}: \operatorname{Mat}_{n}(\mathbb{C}) \rightarrow \operatorname{Mat}_{n}(\mathbb{C})$,

$$X \mapsto A X-X A$$

(i) Compute a basis of eigenvectors, and their associated eigenvalues, when $A$ is the diagonal matrix

$$A=\left(\begin{array}{llll} 1 & & & \\ & 2 & & \\ & & \ddots & \\ & & & n \end{array}\right)$$

What is the rank of $\varphi_{A}$ ?

(ii) Now let $A=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$. Write down the matrix of the linear transformation $\varphi_{A}$ with respect to the standard basis of $\operatorname{Mat}_{2}(\mathbb{C})$.

What is its Jordan normal form?