In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.

In this question, $\lambda = \alpha + \mathrm{i}\beta$, with $(\alpha, \beta)$ in $\mathbb{R}^2$, is a non-real eigenvalue of $M$ and $Z$ in $\mathcal{M}_{n,1}(\mathbb{C})$, non-zero, is such that $MZ = \lambda Z$.

If $M = (m_{i,j})_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ we denote $\bar{M} = (m_{i,j}')_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ with $m_{i,j}' = \bar{m}_{i,j}$ (conjugate of the complex number $m_{i,j}$) for all $(i,j)$ in $\llbracket 1, n \rrbracket^2$ and if $Z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}$ we denote $\bar{Z} = \begin{pmatrix} z_1' \\ \vdots \\ z_n' \end{pmatrix}$ with $z_i' = \bar{z}_i$ for all $i$ in $\llbracket 1, n \rrbracket$.

We set $X = \frac{1}{2}(Z + \bar{Z})$ and $Y = \frac{1}{2\mathrm{i}}(Z - \bar{Z})$.

\textbf{IV.C.1)} Verify that $X$ and $Y$ are in $E$ and show that the family $(X, Y)$ is free in $E$.

\textbf{IV.C.2)} Show that the vector plane $F$ generated by $X$ and $Y$ is stable by $f$ and give the matrix of $f_F$ in the basis $(X, Y)$.