grandes-ecoles 2022 Q45

grandes-ecoles · France · centrale-maths1__official Groups Symplectic and Orthogonal Group Properties

Let $r > 0$ be 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 let $U$ be the matrix of $u$. Given that every complex eigenvalue $\lambda$ of $U$ satisfies $|\lambda| \leq r$, deduce that $1 \leqslant r$.

Let $r > 0$ be 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 let $U$ be the matrix of $u$. Given that every complex eigenvalue $\lambda$ of $U$ satisfies $|\lambda| \leq r$, deduce that $1 \leqslant r$.

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