Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$. We assume that $G$ satisfies hypothesis (H5): $\operatorname{dim}(G) = 2m-2$. Show that if $(w_1, w_2)$ is a characterizing pair of $G$ then $(T(w_1), T(w_2))$ constitutes a basis of $G^\perp$.
Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We assume that $G$ satisfies hypothesis (H5): $\operatorname{dim}(G) = 2m-2$.
Show that if $(w_1, w_2)$ is a characterizing pair of $G$ then $(T(w_1), T(w_2))$ constitutes a basis of $G^\perp$.