Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. 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 $\tau _ { y - x } ^ { \lambda } ( x ) = y$, where $\tau_a^{\lambda}(x) = x + \lambda \omega(a,x)a$.

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 $\tau _ { y - x } ^ { \lambda } ( x ) = y$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. 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 $\omega ( y , z ) \neq 0$.

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 $\omega ( y , z ) \neq 0$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. 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 $\gamma ( x ) = y$.

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 $\gamma ( x ) = y$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and 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 $\omega \left( e _ { 1 } , f _ { 1 } \right) = 1$.

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 $\omega \left( e _ { 1 } , f _ { 1 } \right) = 1$.

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 $u \in \operatorname { Symp } _ { \omega } ( E )$ 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 $\delta _ { 1 } \left( u \left( e _ { 1 } \right) \right) = e _ { 1 }$?

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1, f_1 \in E$ with $\omega(e_1, f_1) = 1$, and let $\delta_1$ be 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.

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.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $P = \operatorname { Vect } \left( e _ { 1 } , f _ { 1 } \right)$ where $\omega(e_1, f_1) = 1$. Let $\delta = \delta_2 \circ \delta_1$ be a composition of at most four symplectic transvections satisfying $\delta(u(e_1)) = e_1$ and $\delta(u(f_1)) = f_1$. Set $v = \delta \circ u$. Show that $P$ is stable under $v$ and determine $v _ { P }$, the endomorphism induced by $v$ on $P$.

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 $P$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $P = \operatorname { Vect} ( e_1, f_1 )$ with $\omega(e_1,f_1)=1$, and $v = \delta \circ u$ where $u \in \operatorname{Symp}_{\omega}(E)$ and $\delta$ is a composition of symplectic transvections with $v|_P = \mathrm{id}_P$. Show that $P ^ { \omega }$ is stable under $v$.

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 $v$.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $P = \operatorname{Vect}(e_1, f_1)$ with $\omega(e_1,f_1)=1$, and $v = \delta \circ u$ where $u \in \operatorname{Symp}_{\omega}(E)$ and $\delta$ is a composition of symplectic transvections with $v|_P = \mathrm{id}_P$. 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.

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.

Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Using the results of questions 34--39, 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 $u = \tau _ { p } \circ \cdots \circ \tau _ { 2 } \circ \tau _ { 1 }$.

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 $u = \tau _ { p } \circ \cdots \circ \tau _ { 2 } \circ \tau _ { 1 }$.

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.

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.

Use the results of subsection III.D to prove the inclusion $\mathrm { Sp } _ { n } ( \mathbb { R } ) \subset \mathrm { SL } _ { n } ( \mathbb { R } )$.

Use the results of subsection III.D to prove the inclusion $\mathrm { Sp } _ { n } ( \mathbb { R } ) \subset \mathrm { SL } _ { n } ( \mathbb { R } )$.

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 $u \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$.