Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $i$ be an integer $\geqslant 1$.
Show that
$$B^i = \begin{pmatrix} \left(\lambda_1 I_{n_1} + J_{n_1}\right)^i & 0 & \cdots & 0 \\ 0 & \left(\lambda_2 I_{n_2} + J_{n_2}\right)^i & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \left(\lambda_k I_{n_k} + J_{n_k}\right)^i \end{pmatrix}$$