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Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2022 centrale-maths1__official

48 maths questions

Q1 Matrices Matrix Algebra and Product Properties View
Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that
$$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$
Show that $A = B$.
Q2 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.
Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.
Q3 Proof Direct Proof of a Stated Identity or Equality View
Show that, if $\omega$ is a symplectic form on $E$, then for every vector $x$ in $E$, $\omega ( x , x ) = 0$.
Q5 Proof True/False Justification View
Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, with $\omega$-orthogonal $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?
Q6 Proof Proof That a Map Has a Specific Property View
For every $x \in E$, we denote by $\omega ( x , \cdot )$ the linear map from $E$ to $\mathbb { R }$, $y \mapsto \omega ( x , y )$ and we consider
$$d _ { \omega } : \left\lvert \, \begin{array} { c c c } E & \rightarrow & \mathcal { L } ( E , \mathbb { R } ) \\ x & \mapsto & \omega ( x , \cdot ) \end{array} \right.$$
Show that $d _ { \omega }$ is an isomorphism.
Q7 Proof Proof That a Map Has a Specific Property View
For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.
Q8 Matrices Linear Transformation and Endomorphism Properties View
Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.
Q9 Groups Symplectic and Orthogonal Group Properties View
Show that the restriction $\omega _ { F }$ of $\omega$ to $F ^ { 2 }$ defines a symplectic form on $F$ if and only if $F \oplus F ^ { \omega } = E$.
Q10 Matrices Bilinear and Symplectic Form Properties View
We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.
Q11 Matrices Determinant and Rank Computation View
We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$ satisfying $\omega(x,y) = X^{\top} \Omega Y$ for all $x,y \in \mathbb{R}^n$. Deduce that $\Omega$ is antisymmetric and invertible.
Q12 Matrices Determinant and Rank Computation View
Given that any symplectic form on $\mathbb{R}^n$ has an associated matrix $\Omega$ that is antisymmetric and invertible, conclude that the integer $n$ is even.
Q13 Matrices Bilinear and Symplectic Form Properties View
We assume $n = 2m$ and denote by $J \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined in blocks by
$$J = \left( \begin{array} { c c } 0 & - I _ { m } \\ I _ { m } & 0 \end{array} \right)$$
and we denote by $j$ the endomorphism of $\mathbb { R } ^ { n }$ canonically associated with $J$. Show that the map $b _ { s } : \left\lvert \, \begin{array} { c c l } \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( x , y ) & \mapsto & \langle x , j ( y ) \rangle \end{array} \right.$ is a symplectic form on $\mathbb { R } ^ { n }$.
Q14 Proof Direct Proof of a Stated Identity or Equality View
Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $\lambda , \mu$ be real eigenvalues of $u$, and let $E _ { \lambda } ( u ) , E _ { \mu } ( u )$ be the associated eigenspaces. Show that, if $\lambda \mu \neq 1$, then the subspaces $E _ { \lambda } ( u )$ and $E _ { \mu } ( u )$ are $\omega$-orthogonal, that is:
$$\forall x \in E _ { \lambda } ( u ) , \quad \forall y \in E _ { \mu } ( u ) , \quad \omega ( x , y ) = 0$$
Q15 Proof Proof of Equivalence or Logical Relationship Between Conditions View
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 $\mathbb { R } ^ { n }$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.
Q16 Groups Symplectic and Orthogonal Group Properties View
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 $J$.
Q17 Matrices Bilinear and Symplectic Form Properties View
Let $A , B , C , D$ be in $\mathcal { M } _ { m } ( \mathbb { R } )$ and let $M = \left( \begin{array} { l l } A & B \\ C & D \end{array} \right) \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ (block decomposition). Show that $M \in \mathrm { Sp } _ { 2 m } ( \mathbb { R } )$ if and only if
$$A ^ { \top } C \text { and } B ^ { \top } D \text { are symmetric } \quad \text { and } \quad A ^ { \top } D - C ^ { \top } B = I _ { m } .$$
Q18 Matrices Matrix Group and Subgroup Structure View
Show that $\mathrm { Sp } _ { 2 } ( \mathbb { R } ) = \mathrm { SL } _ { 2 } ( \mathbb { R } )$.
Q19 Matrices Structured Matrix Characterization View
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$,
$$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) .$$
Q20 3x3 Matrices Block Matrix Multiplication and Determinant Identity View
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J$, and it has been shown that $M \in \mathcal{C}_J$ if and only if $M = \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right)$ for some $U, V \in \mathcal{M}_m(\mathbb{R})$. Deduce that, for every matrix $M \in \mathcal { C } _ { J } , \operatorname { det } ( M ) \geqslant 0$.
One may consider the product of block matrices $\left( \begin{array} { c c } I _ { m } & 0 \\ \mathrm { i } I _ { m } & I _ { m } \end{array} \right) \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) \left( \begin{array} { c c } I _ { m } & 0 \\ - \mathrm { i } I _ { m } & I _ { m } \end{array} \right)$.
Q21 Groups Symplectic and Orthogonal Group Properties View
We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$. We equip $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space of finite dimension. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } )$ is a compact subgroup of the symplectic group $\operatorname { Sp } _ { n } ( \mathbb { R } )$.
Q22 Matrices Structured Matrix Characterization View
We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices, and $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of $J$. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$.
Q23 Matrices Determinant and Rank Computation View
Given that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$ and that for every matrix $M \in \mathcal { C } _ { J }$, $\det(M) \geq 0$, deduce that, for every matrix $M$ in $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.
Q24 Matrices Bilinear and Symplectic Form Properties View
We consider a matrix $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$. Let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.
Q25 Matrices Matrix Group and Subgroup Structure View
We consider a matrix $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ a symmetric symplectic matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Justify that $S$ is invertible then show that the matrix $O$ defined by $O = M S ^ { - 1 }$ belongs to the group $\mathrm { OSp } _ { n } ( \mathbb { R } )$.
Q26 Matrices Determinant and Rank Computation View
Using the polar decomposition $M = OS$ where $O \in \operatorname{OSp}_n(\mathbb{R})$ and $S$ is a symmetric symplectic matrix with strictly positive eigenvalues, conclude that the determinant of the matrix $M \in \mathrm{Sp}_n(\mathbb{R})$ is equal to 1.
Q27 Matrices Bilinear and Symplectic Form Properties View
Let $(E, \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.
Q28 Groups Symplectic and Orthogonal Group Properties View
Let $a \in E$ be a non-zero vector and let $\lambda$ and $\mu$ be real numbers. Show that $\tau _ { a } ^ { \mu } \circ \tau _ { a } ^ { \lambda } = \tau _ { a } ^ { \lambda + \mu }$.
Q29 Groups Symplectic and Orthogonal Group Properties View
Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.
Q30 Groups Symplectic and Orthogonal Group Properties View
Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. Is the inverse $\left( \tau _ { a } ^ { \lambda } \right) ^ { - 1 }$ still a symplectic transvection?
Q31 Groups Symplectic and Orthogonal Group Properties View
We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) \neq 0$. Show that there exists $\lambda \in \mathbb { R }$ such that $\tau _ { y - x } ^ { \lambda } ( x ) = y$.
Q32 Groups Symplectic and Orthogonal Group Properties View
We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) = 0$. Show that there exists a vector $z \in E$ such that $\omega ( x , z ) \neq 0$ and $\omega ( y , z ) \neq 0$.
Q33 Groups Symplectic and Orthogonal Group Properties View
Prove the following lemma: For all non-zero vectors $x$ and $y$ of $E$, there exists a composition $\gamma$ of at most two symplectic transvections of $E$ such that $\gamma ( x ) = y$.
Q34 Groups Symplectic and Orthogonal Group Properties View
Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$. Let $e _ { 1 } \in E$ be a non-zero vector. Justify the existence of $f _ { 1 } \in E$, not collinear with $e _ { 1 }$, such that $\omega \left( e _ { 1 } , f _ { 1 } \right) = 1$.
Q35 Groups Symplectic and Orthogonal Group Properties View
Let $u \in \operatorname { Symp } _ { \omega } ( E )$ be a symplectic endomorphism of $E$, and 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 $\delta _ { 1 } \left( u \left( e _ { 1 } \right) \right) = e _ { 1 }$?
Q36 Groups Symplectic and Orthogonal Group Properties View
Let $u \in \operatorname { Symp } _ { \omega } ( E )$, $e_1 \in E$ non-zero, $f_1 \in E$ not collinear with $e_1$ such that $\omega(e_1, f_1) = 1$, and $\delta_1$ a composition of at most two symplectic transvections such that $\delta_1(u(e_1)) = e_1$. Let $\tilde { f } _ { 1 }$ denote the vector $\delta _ { 1 } \left( u \left( f _ { 1 } \right) \right)$. Show that there exists a composition $\delta _ { 2 }$ of at most two symplectic transvections of $E$ such that
$$\left\{ \begin{array} { l } \delta _ { 2 } \left( e _ { 1 } \right) = e _ { 1 } \\ \delta _ { 2 } \left( \tilde { f } _ { 1 } \right) = f _ { 1 } \end{array} \right.$$
One may adapt the proof of the preceding lemma.
Q37 Groups Symplectic and Orthogonal Group Properties View
Let $u \in \operatorname{Symp}_\omega(E)$, $P = \operatorname{Vect}(e_1, f_1)$ with $\omega(e_1, f_1) = 1$, and $v = \delta \circ u$ where $\delta = \delta_2 \circ \delta_1$ satisfies $\delta(u(e_1)) = e_1$ and $\delta(u(f_1)) = f_1$. Show that $P$ is stable under $v$ and determine $v _ { P }$, the endomorphism induced by $v$ on $P$.
Q38 Groups Symplectic and Orthogonal Group Properties View
With the same setup as Q37 ($v = \delta \circ u$, $P = \operatorname{Vect}(e_1, f_1)$, $P^\omega$ the $\omega$-orthogonal of $P$), show that $P ^ { \omega }$ is stable under $v$.
Q39 Groups Symplectic and Orthogonal Group Properties View
With the same setup as Q37--Q38, show that the restriction $\omega _ { P ^ { \omega } }$ of $\omega$ to $P ^ { \omega } \times P ^ { \omega }$ equips $P ^ { \omega }$ with a symplectic space structure and that the endomorphism $v _ { P ^ { \omega } }$ induced by $v$ on $P ^ { \omega }$ is a symplectic endomorphism.
Q40 Groups Symplectic and Orthogonal Group Properties View
Using the results of Q34--Q39, prove the following theorem: Every symplectic endomorphism of $E$ can be written as the composition of at most $2n = 4m$ symplectic transvections of $E$: if $u \in \operatorname { Symp } _ { \omega } ( E )$, there exist an integer $p \leqslant 4 m$ and $\tau _ { 1 } , \tau _ { 2 } , \ldots , \tau _ { p }$ symplectic transvections of $E$ such that $u = \tau _ { p } \circ \cdots \circ \tau _ { 2 } \circ \tau _ { 1 }$.
Q41 Groups Symplectic and Orthogonal Group Properties View
We equip the space $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space. Show that the symplectic group $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is an arc-connected subset of this space.
Q42 Groups Symplectic and Orthogonal Group Properties View
Use the results of subsection III.D to prove the inclusion $\mathrm { Sp } _ { n } ( \mathbb { R } ) \subset \mathrm { SL } _ { n } ( \mathbb { R } )$.
Q43 Groups Symplectic and Orthogonal Group Properties View
We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $$B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$$ and the symplectic cylinder of radius $r$ is $$Z ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant r ^ { 2 } \right\}.$$ Show that, for all $r > 0$, there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$.
Q44 Groups Symplectic and Orthogonal Group Properties View
Let $r > 0$ be such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$. Let $U \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ denote the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { 2 m }$. Let $\lambda \in \mathbb { C }$ be a complex eigenvalue of the matrix $U$. Show that $| \lambda | \leqslant r$.
For the case $\lambda$ non-real, if $P$ and $Q$ in $\mathcal { M } _ { 2 m , 1 } ( \mathbb { R } )$ are such that $Z = P + \mathrm { i } Q$ is an eigenvector column of $U$ for the eigenvalue $\lambda$, one may show that $\| U P \| ^ { 2 } + \| U Q \| ^ { 2 } = | \lambda | ^ { 2 } \left( \| P \| ^ { 2 } + \| Q \| ^ { 2 } \right)$.
Q45 Groups Symplectic and Orthogonal Group Properties View
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 $1 \leqslant r$.
Q46 Groups Symplectic and Orthogonal Group Properties View
Under what necessary and sufficient condition on $r > 0$ does there exist $u$ belonging to $\mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$?
Q47 Groups Symplectic and Orthogonal Group Properties View
Let $r > 0$ be such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$. We denote by $M \in \operatorname { Sp } _ { 2 m } ( \mathbb { R } )$ the matrix of $\psi$ in the canonical basis $\left( e _ { 1 } , \ldots , e _ { m } , f _ { 1 } , \ldots , f _ { m } \right)$ of $\mathbb { R } ^ { 2 m }$ and by $\psi ^ { \top }$ the endomorphism canonically associated with $M ^ { \top }$. Show that $\left| b _ { s } \left( \psi ^ { \top } \left( e _ { 1 } \right) , \psi ^ { \top } \left( f _ { 1 } \right) \right) \right| = 1$ then that $\left\| \psi ^ { \top } \left( e _ { 1 } \right) \right\| \geqslant 1$ or $\left\| \psi ^ { \top } \left( f _ { 1 } \right) \right\| \geqslant 1$.
Q48 Groups Symplectic and Orthogonal Group Properties View
Let $r > 0$ be such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$. Using the result of Q47, show that $1 \leqslant r$.
Q49 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.