In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$. Let $F \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by $$[F]_{k,j} = \frac{1}{\sqrt{n}} \omega^{(k-1)(j-1)} \text{ for all } (k,j) \in \llbracket 1; n \rrbracket^2,$$ where $\omega = e^{-2\pi i/n}$ (here $i$ denotes the usual complex number satisfying $i^2 = -1$). (a) Show that $F$ is invertible and that $F^{-1} = \bar{F}$. (b) Show that $F^2 \in \mathscr{M}_n(\mathbb{R})$. (c) Deduce that $F^4 = I_n$ and that $F \in \mathbb{M}_n(u)$. (d) Deduce that $$\begin{aligned}
u(F) = & \frac{1}{4}\left(U(1)(F + I_n) - U(-1)(F - I_n)\right)(F^2 + I_n) \\
& + \frac{i}{4}\left(U(i)(F + iI_n) - U(-i)(F - iI_n)\right)(F^2 - I_n)
\end{aligned}$$
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Let $F \in \mathscr{M}_n(\mathbb{C})$ be the matrix defined by
$$[F]_{k,j} = \frac{1}{\sqrt{n}} \omega^{(k-1)(j-1)} \text{ for all } (k,j) \in \llbracket 1; n \rrbracket^2,$$
where $\omega = e^{-2\pi i/n}$ (here $i$ denotes the usual complex number satisfying $i^2 = -1$).\\
(a) Show that $F$ is invertible and that $F^{-1} = \bar{F}$.\\
(b) Show that $F^2 \in \mathscr{M}_n(\mathbb{R})$.\\
(c) Deduce that $F^4 = I_n$ and that $F \in \mathbb{M}_n(u)$.\\
(d) Deduce that
$$\begin{aligned}
u(F) = & \frac{1}{4}\left(U(1)(F + I_n) - U(-1)(F - I_n)\right)(F^2 + I_n) \\
& + \frac{i}{4}\left(U(i)(F + iI_n) - U(-i)(F - iI_n)\right)(F^2 - I_n)
\end{aligned}$$