# Definition The $2n \times 2n$ symplectic matrix is defined by: $ \mathbb{J} \equiv \begin{pmatrix} \mathbb{0} & \mathbb{1} \\ -\mathbb{1} & \mathbb{0} \end{pmatrix} $ where $\mathbb{1}$ is the $n\times n$ identity matrix, $\mathbb{0}$ is the $n \times n$ zero [[matrix]].