symplectic matrix
A real $2n\times 2n$ matrix $A\in {\mathrm{M}}_{2n}(\mathbb{R})$ is a symplectic matrix if $AJ{A}^{T}=J$, where ${A}^{T}$ is the transpose^{} of $A$, and $J\in \mathrm{O}(2n)$ is the orthogonal matrix^{}
$$J=\left(\begin{array}{cc}\hfill \mathrm{\U0001d7ce}\hfill & \hfill {\mathbf{I}}_{n}\hfill \\ \hfill {\mathbf{I}}_{n}\hfill & \hfill \mathrm{\U0001d7ce}\hfill \end{array}\right).$$ 
Here ${\mathbf{I}}_{n}\in {\mathrm{M}}_{n}(\mathbb{R})$ is the identity $n\times n$ matrix and $\mathrm{\U0001d7ce}\in {\mathrm{M}}_{n}(\mathbb{R})$ 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 $\mathrm{\Psi}=\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right)$, where $A,B,C,D$ are $n\times n$ matrices. Then $\mathrm{\Psi}$ is symplectic if and only if
$$A{D}^{T}B{C}^{T}=I,A{B}^{T}=B{A}^{T},C{D}^{T}=D{C}^{T}.$$ 
4.
If $X$ and $Y$ are real $n\times n$ matrices, then $U=X+iY$ is unitary if and only if $\left(\begin{array}{cc}\hfill X\hfill & \hfill Y\hfill \\ \hfill Y\hfill & \hfill X\hfill \end{array}\right)$ is symplectic.
Title  symplectic matrix 

Canonical name  SymplecticMatrix 
Date of creation  20130322 13:32:28 
Last modified on  20130322 13:32:28 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  11 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 53D05 