Define a Jordan block $J_{m}(\lambda)$. What does it mean for a complex $n \times n$ matrix to be in Jordan normal form?

If $A$ is a matrix in Jordan normal form for an endomorphism $\alpha: V \rightarrow V$, prove that

$$\operatorname{dim} \operatorname{Ker}\left((\alpha-\lambda I)^{r}\right)-\operatorname{dim} \operatorname{Ker}\left((\alpha-\lambda I)^{r-1}\right)$$

is the number of Jordan blocks $J_{m}(\lambda)$ of $A$ with $m \geqslant r$.

Find a matrix in Jordan normal form for $J_{m}(\lambda)^{2}$. [Consider all possible values of $\lambda$.]

Find a matrix in Jordan normal form for the complex matrix

$$\left[\begin{array}{cccc} 0 & 0 & 0 & a_{1} \\ 0 & 0 & a_{2} & 0 \\ 0 & a_{3} & 0 & 0 \\ a_{4} & 0 & 0 & 0 \end{array}\right]$$

assuming it is invertible.