A transformation matrix T with an even number of rows and columns is called symplectic iff T^T⋅J⋅T=J.

(with T^T being the transposed matrix T)

(with J:=\begin{pmatrix} 0 & E⃗_n \\ -E⃗_n & 0 \end{pmatrix}).

(with E_n being the unit vector).

The set of all symplectic matrices is a group, that is, has addition and multiplication and inversion with the usual properties.

Note that:

\det T=1
J^T=-J
J^{-1}=-J
J⋅J=-E_n