Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$ and $S \in \mathrm{Sp}_n(\mathbb{R})$. Justify that $S$ is invertible then show that the matrix $O$ defined by $O = M S ^ { - 1 }$ belongs to the group $\mathrm { OSp } _ { n } ( \mathbb { R } )$.

Show that $\mathrm { Sp } _ { 2 } ( \mathbb { R } ) = \mathrm { SL } _ { 2 } ( \mathbb { R } )$.

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 $\mathrm { OSp } _ { n } ( \mathbb { R } )$.

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Verify that $s_{w_1}$, $s_{w_2}$ and $s_{w_3}$ belong to $\Gamma$ and calculate the corresponding matrices.

Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is closed under multiplication.

Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is stable under multiplication.