# Generators of the symplectic group

Let $$n$$ be an integer $$\ge 1$$. We define the $$2n\times 2n$$ matrix $$\sigma$$ with $$n\times n$$ blocks by $$\sigma=\begin{pmatrix}0&I_n \\-I_n&0\end{pmatrix}.$$ The symplectic group $$Sp(n)$$ is defined as the set of $$2n\times 2n$$ matrices $$S$$ such that $$S^*\sigma S=\sigma.$$ It is a classical fact that the symplectic group is a subgroup of $$Sl(2n, \mathbb R)$$ and is generated by matrices $$\begin{pmatrix}B^{-1}&0 \\0&B^*\end{pmatrix}\text{(with B\in Gl(n,\mathbb R)}),\quad \begin{pmatrix}I_n&0 \\A&I_n\end{pmatrix}\text{(with A symmetric n\times n}) \tag{\ast}$$ and linear mapping $$M_j, 1\le j\le n$$ such that $$(x_j, \xi_j)\mapsto (\xi_j, -x_j),\ \text {other coordinates fixed}.$$ I claim that it is enough to get a set of generators to consider ($$\ast$$) along with $$\begin{pmatrix}I_n&C \\0&I_n\end{pmatrix}\text{(with C symmetric n\times n }), \text{as well as \sigma}.$$ It should be classical. I would be grateful for a reference in the literature.