(i) Show that two $n \times n$ complex matrices $A, B$ are similar (i.e. there exists invertible $P$ with $A=P^{-1} B P$ ) if and only if they represent the same linear map $\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ with respect to different bases.

(ii) Explain the notion of Jordan normal form of a square complex matrix.

(iii) Show that any square complex matrix $A$ is similar to its transpose.

(iv) If $A$ is invertible, describe the Jordan normal form of $A^{-1}$ in terms of that of $A$.

Justify your answers.