PlanetMath: symplectic matrix

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

symplectic matrix

(Definition)

A real $2n\times 2n$ matrix $A\in\mathrm{M}_{2n}(\mathbb{R})$ is a symplectic matrix if , where is the transpose of , and $J\in\mathrm{O}(2n)$ is the orthogonal matrix

$\displaystyle J=\left( \begin{array}{cc} \mathbf 0 & \mathbf{I}_n \ -\mathbf{I}_n & \mathbf 0 \end{array} \right).$

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:

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