grandes-ecoles 2022 Q35

grandes-ecoles · France · centrale-maths1__mp Groups Symplectic and Orthogonal Group Properties

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1 \in E$ be a non-zero vector. Why does there exist a composition $\delta _ { 1 }$ of at most two symplectic transvections of $E$ such that $\delta _ { 1 } \left( u \left( e _ { 1 } \right) \right) = e _ { 1 }$?

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1 \in E$ be a non-zero vector. Why does there exist a composition $\delta _ { 1 }$ of at most two symplectic transvections of $E$ such that $\delta _ { 1 } \left( u \left( e _ { 1 } \right) \right) = e _ { 1 }$?

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