Let $F$ be a field. For an integer $n \ge 1$, and ordered partition of $n$ is a sequence $\underline{r} = \{r_1, \dots, r_m\}$ of positive integers such that $r_1 \le \dots \le r_m$ and $\sum r_j = n$. For each such $\underline{r}$, define $N(\underline{R}) \in \text{Mat}_{n \times n}(F)$ to be the block matrix given by $r_j \times r_j$ nilpotent Jordan blocks $($i.e., $1$'s just below main diagonal, $0$'s elsewhere$)$, with blocks stacked along the main diagonal in order of increasing size.
- Prove that $N(\underline{r})$ is nilpotent and compute its rational canonical form.
- Use Jordan canonical form to prove that every nilpotent $N \in \text{Mat}_{n \times n}(F)$ is $\text{GL}_n(F)$-conjugate to $N(\underline{r})$ for a unique $\underline{R}$.
I understand that for the second part we want to show $N(\underline{r}) = MNM^{-1}$ for some $M \in \text{GL}_n(F)$. I also know that Jordan canonical form only requires the minimal polynomial to split into linears in $F[t]$, which is an automatic condition when $F$ is algebraically closed, and also when the linear map is nilpotent.