We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$ We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the set $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$. Let $u \in \mathcal{W}$. Show that $(\bar{u})^{k}(z) = \pi(u^{k}(z))$ for every $k \in \mathbf{N}$ and every $z \in H$. Deduce that $\overline{\mathcal{V}}$ is a nilpotent vector subspace of $\mathcal{L}(H)$.
We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the set $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$.
Let $u \in \mathcal{W}$. Show that $(\bar{u})^{k}(z) = \pi(u^{k}(z))$ for every $k \in \mathbf{N}$ and every $z \in H$. Deduce that $\overline{\mathcal{V}}$ is a nilpotent vector subspace of $\mathcal{L}(H)$.