Symplectic group

Description

Let $\mathbb{K}$ be a field, let $n \ge 1$, and let $$\Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix},$$ a $2n \times 2n$ block matrix, where $I_n$ is the $n \times n$ identity matrix. The symplectic group is $$\operatorname{Sp}(2n, \mathbb{K}) = \set{M \in \operatorname{GL}(2n, \mathbb{K}) : M^T \Omega M = \Omega}.$$ It is a group under matrix multiplication, a subgroup of the general linear group. Check: $I^T \Omega I = \Omega$; if $M^T \Omega M = \Omega$ and $N^T \Omega N = \Omega$, then $(MN)^T \Omega (MN) = N^T (M^T \Omega M) N = \Omega$; and multiplying $M^T \Omega M = \Omega$ by $(M^T)^{-1}$ on the left and $M^{-1}$ on the right gives $(M^{-1})^T \Omega M^{-1} = \Omega$.