grandes-ecoles 2022 Q45

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

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$, and every complex eigenvalue $\lambda$ of the matrix $U$ of $u$ satisfies $|\lambda| \leq r$. Deduce that $1 \leqslant r$.

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$, and every complex eigenvalue $\lambda$ of the matrix $U$ of $u$ satisfies $|\lambda| \leq r$. Deduce that $1 \leqslant r$.

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