Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Justify that if $\lambda \neq -1$, $F$ is a vector space of dimension 4. Show that, in this case, $$\left(X,\ \frac{-1}{\sqrt{-\lambda}} MX,\ -J_{n} X,\ \frac{1}{\sqrt{-\lambda}} J_{n} MX\right)$$ is an orthonormal basis of $F$. Then give the matrix of the application $m_{F}$ induced by $m$ on $F$ in the basis obtained.
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Let $X$ denote an eigenvector of $M^{2}$ of norm 1 associated with a certain eigenvalue $\lambda$, and let $F = \operatorname{Vect}(X, MX, J_{n} X, J_{n} MX)$. Justify that if $\lambda \neq -1$, $F$ is a vector space of dimension 4. Show that, in this case,
$$\left(X,\ \frac{-1}{\sqrt{-\lambda}} MX,\ -J_{n} X,\ \frac{1}{\sqrt{-\lambda}} J_{n} MX\right)$$
is an orthonormal basis of $F$. Then give the matrix of the application $m_{F}$ induced by $m$ on $F$ in the basis obtained.