# Definition (Vector Spaces) Given a finite-[[Dimension|dimensional]] [[Symplectic Vector Space]] $(Z, \Omega)$, the set of all [[Symplectomorphism|symplectomorphisms]] forms a [[group]] called the *symplectic group*, denoted by $Sp(Z, \Omega)$. In a canonical basis, a matrix $A$ is symplectic if and only if: $ A^T \mathbb{J} A = \mathbb{J} $ where $\mathbb{J}$ is the [[Symplectic Matrix]]. Assume we have $Z = W \times W^*$ and a canonical basis, then if A has the matrix form: $ A=\left[\begin{array}{ll} A_{q q} & A_{q p} \\ A_{p q} & A_{p p} \end{array}\right] $ then we have either of the two following conditions: 1. $A_{qq}A_{qp}^T$ and $A_{pp}A_{pq}^T$ are symmetric and $A_{qq}A_{pp}^T - A_{qp}A_{pq}^T = \mathbb{1}$ 2. $A_{pq}^TA_{qq}$ and $A_{qp}^TA_{pp}$ are symmetric and $A_{qq}^TA_{pp} - A_{pq}^TA_{qp} = \mathbb{1}$ In infinite dimensions, $Sp(Z, \Omega) \subset GL(Z)$ is the subset of the [[general linear group]] that leaves $\Omega$ fixed.