# symplectic matrix

A real $2n\times 2n$ matrix $A\in\mathrm{M}_{2n}(\mathbb{R})$ is a symplectic matrix if $AJA^{T}=J$, where $A^{T}$ is the transpose  of $A$, and $J\in\mathrm{O}(2n)$ is the orthogonal matrix  $J=\left(\begin{array}[]{cc}\mathbf{0}&\mathbf{I}_{n}\\ -\mathbf{I}_{n}&\mathbf{0}\end{array}\right).$

Here $\mathbf{I}_{n}\in\mathrm{M}_{n}(\mathbb{R})$ is the identity $n\times n$ matrix and $\mathbf{0}\in\mathrm{M}_{n}(\mathbb{R})$ is the zero $n\times n$ matrix.

Symplectic matrices satisfy the following properties:

1. 1.
2. 2.

With standard matrix multiplication, symplectic $2n\times 2n$ matrices form a group denoted by $\mathrm{Sp}(2n)$.

3. 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

 $AD^{T}-BC^{T}=I,\,\,\,\,\,AB^{T}=BA^{T},\,\,\,\,\,CD^{T}=DC^{T}.$
4. 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.

Title symplectic matrix SymplecticMatrix 2013-03-22 13:32:28 2013-03-22 13:32:28 matte (1858) matte (1858) 11 matte (1858) Definition msc 53D05