Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We consider the element $G_a$ of $\mathcal{L}(W_{\ell})$ whose matrix in the basis $\{v_i\}_{0 \leq i < \ell}$ is:
$$\left(\begin{array}{cccccc}
0 & 0 & 0 & \cdots & 0 & a \\
1 & 0 & 0 & \cdots & 0 & 0 \\
0 & 1 & 0 & \cdots & 0 & 0 \\
\vdots & 0 & \ddots & \ddots & \vdots & \vdots \\
0 & \vdots & \ddots & \ddots & 0 & 0 \\
0 & 0 & \ldots & 0 & 1 & 0
\end{array}\right)$$
10a. Calculate $G_a^{\ell}$. Show that $G_a$ is diagonalizable.
10b. Let $b$ be an $\ell$-th root of $a$. Calculate the eigenvectors of $G_a$ and the associated eigenvalues in terms of $b, q$ and the $v_i$.