grandes-ecoles 2022 Q16

grandes-ecoles · France · centrale-maths1__official Groups Symplectic and Orthogonal Group Properties
We denote by $\mathrm { Sp } _ { n } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$ the set of real symplectic matrices. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$. 
We denote by $\mathrm { Sp } _ { n } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$ the set of real symplectic matrices. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$.
Paper Questions
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