Editing Symplectic matrix (section)

==Properties==

=== 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>.

=== Inverse matrix ===
Every symplectic matrix is invertible with the [[inverse matrix]] given by
<math display="block">
M^{-1} = \Omega^{-1} M^\text{T} \Omega.
</math>
Furthermore, the [[matrix multiplication|product]] of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a [[group (mathematics)|group]]. There exists a natural [[manifold]] structure on this group which makes it into a (real or complex) [[Lie group]] called the [[symplectic group]].

=== Determinantal properties ===
It follows easily from the definition that the [[determinant]] of any symplectic matrix is ±1. Actually, it turns out that the determinant is always +1 for any field. One way to see this is through the use of the [[Pfaffian]] and the identity
<math display="block">\mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).</math>
Since <math>M^\text{T} \Omega M = \Omega</math> and <math>\mbox{Pf}(\Omega) \neq 0</math> we have that <math>\det(M) = 1</math>.

When the underlying field is real or complex, one can also show this by factoring the inequality <math>\det(M^\text{T} M + I) \ge 1</math>.<ref>{{cite journal |last=Rim |first=Donsub |date=2017 |title=An elementary proof that symplectic matrices have determinant one |journal=Adv. Dyn. Syst. Appl. |volume=12 |issue=1 |pages=15–20 |doi=10.37622/ADSA/12.1.2017.15-20 |arxiv=1505.04240  |s2cid=119595767 }}</ref>

=== Block form of symplectic matrices ===
Suppose Ω is given in the standard form and let <math>M</math> be a <math>2n\times 2n</math> [[block matrix]] given by
<math display="block">M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}</math>
where <math>A,B,C,D</math> are <math>n\times n</math> matrices. The condition for <math>M</math> to be symplectic is equivalent to the two following equivalent conditions<ref>{{cite web|last1=de Gosson|first1=Maurice|title=Introduction to Symplectic Mechanics: Lectures I-II-III|url=https://www.ime.usp.br/~piccione/Downloads/LecturesIME.pdf}}</ref><blockquote><math>A^\text{T}C,B^\text{T}D</math> symmetric, and <math>A^\text{T} D - C^\text{T} B = I</math></blockquote><blockquote><math>AB^\text{T},CD^\text{T}</math> symmetric, and <math>AD^\text{T} - BC^\text{T} = I</math></blockquote>The second condition comes from the fact that if <math>M</math> is symplectic, then <math>M^T</math> is also symplectic. When <math>n=1</math> these conditions reduce to the single condition <math>\det(M)=1</math>. Thus a <math>2\times 2</math> matrix is symplectic [[iff]] it has unit determinant.

==== Inverse matrix of block matrix ====
With <math>\Omega</math> in standard form, the inverse of <math>M</math> is given by
<math display="block">
M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}.</math>
The group has dimension <math>n(2n+1)</math>. This can be seen by noting that <math>( M^\text{T} \Omega M)^\text{T} = -M^\text{T} \Omega M</math> is anti-symmetric. Since the space of anti-symmetric matrices has dimension <math>\binom{2n}{2},</math> the identity <math> M^\text{T} \Omega M = \Omega</math> imposes <math>2n \choose 2</math> constraints on the <math>(2n)^2</math> coefficients of <math>M</math> and leaves <math>M</math> with <math>n(2n+1)</math> independent coefficients.