Editing Symplectic matrix (section)

=== Generators for symplectic matrices ===
Every symplectic matrix has determinant <math>+1</math>, and the <math>2n\times 2n</math> symplectic matrices with real entries form a [[subgroup]] of the [[general linear group]] <math>\mathrm{GL}(2n;\mathbb{R})</math> under [[matrix multiplication]] since being symplectic is a property stable under matrix multiplication. [[Topology|Topologically]], this [[symplectic group]] is a [[connected space|connected]] [[compact space|noncompact]] [[real Lie group]] of real dimension <math>n(2n+1)</math>, and is denoted <math>\mathrm{Sp}(2n;\mathbb{R})</math>. The symplectic group can be defined as the set of [[linear transformations]] that preserve the symplectic form of a real [[symplectic vector space]].

This symplectic group has a distinguished [[Generating set of a group|set of generators]], which can be used to find all possible symplectic matrices. This includes the following sets
<math display="block">\begin{align}
D(n) =& \left\{
\begin{pmatrix}
A & 0 \\
0 & (A^T)^{-1}
\end{pmatrix} : A \in \text{GL}(n;\mathbb{R})
\right\} \\
N(n) =& \left\{
\begin{pmatrix}
I_n & B \\
0 & I_n
\end{pmatrix} : B \in \text{Sym}(n;\mathbb{R})
\right\}
\end{align}</math>
where <math>\text{Sym}(n;\mathbb{R})</math> is the set of <math>n\times n</math> [[Symmetric matrix|symmetric matrices]]. Then, <math>\mathrm{Sp}(2n;\mathbb{R})</math> is generated by the set<ref>{{Cite book|last=Habermann, Katharina, 1966-|url=http://worldcat.org/oclc/262692314|title=Introduction to symplectic Dirac operators|date=2006|publisher=Springer|isbn=978-3-540-33421-7|oclc=262692314}}</ref><sup>p. 2</sup>
<math display="block">\{\Omega \} \cup D(n) \cup N(n)</math>
of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in <math>D(n)</math> and <math>N(n)</math> together, along with some power of <math>\Omega</math>.