Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. The standard symplectic form is $b_s(x,y) = \langle x, j(y) \rangle$ where $j$ is canonically associated with $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.
Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. The standard symplectic form is $b_s(x,y) = \langle x, j(y) \rangle$ where $j$ is canonically associated with $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.