A real $2n\times 2n$ matrix $A\in\mm_{2n}(\R)$ is a symplectic matrix if $A J A^T = J$ , where $A^T$ is the transpose of $A$ , and $J\in\oo(2n)$ is the orthogonal matrix $$J=\left( \begin{array}{cc} \mathbf 0 & \I_n \\ -\I_n & \mathbf 0 \end{array} \right).$$ Here $\I_n\in\mm_n(\R)$ is the identity $n\times n$ matrix and $\mathbf 0\in\mm_n(\R)$ is the zero $n\times n$ matrix.

Symplectic matrices satisfy the following properties:

1. The determinant of a symplectic matrix equals one.
2. With standard matrix multiplication, symplectic $2n\times 2n$ matrices form a group denoted by $\mathrm{Sp}(2n)$ .
3. Suppose $\Psi=\begin{pmatrix} A&B \\ C & D \end{pmatrix}$ , where $A,B,C,D$ are $n\times n$ matrices. Then $\Psi$ is symplectic if and only if $$A D^T - BC^T = I, \,\,\,\,\, AB^T=BA^T, \,\,\,\,\, CD^T=DC^T.$$
4. If $X$ and $Y$ are real $n\times n$ matrices, then $U=X+iY$ is unitary if and only if $\begin{pmatrix} X & -Y \\ Y & X \end{pmatrix}$ is symplectic.