Let $d \geqslant 1$, and let $J_{d}=\left(\begin{array}{ccccc}0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ & & \cdots & \cdots & \\ 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \ldots & 0 & 0\end{array}\right) \in \operatorname{Mat}_{d}(\mathbb{C})$.

(a) (i) Compute $J_{d}^{n}$, for all $n \geqslant 0$.

(ii) Hence, or otherwise, compute $\left(\lambda I+J_{d}\right)^{n}$, for all $n \geqslant 0$.

(b) Let $V$ be a finite-dimensional vector space over $\mathbb{C}$, and let $\varphi \in \operatorname{End}(V)$. Suppose $\varphi^{n}=0$ for some $n>1$.

(i) Determine the possible eigenvalues of $\varphi$.

(ii) What are the possible Jordan blocks of $\varphi$ ?

(iii) Show that if $\varphi^{2}=0$, there exists a decomposition

$$V=U \oplus W_{1} \oplus W_{2}$$

where $\varphi(U)=\varphi\left(W_{1}\right)=0, \varphi\left(W_{2}\right)=W_{1}$, and $\operatorname{dim} W_{2}=\operatorname{dim} W_{1}$.