Editing Symplectic matrix (section)

==Diagonalization and decomposition==
{{bullet list
|For any [[Positive-definite matrix|positive definite]] symmetric <math>2n\times 2n</math> real symplectic matrix <math>S</math>, there is a symplectic unitary <math>U</math>, <math display="block">U \in \mathrm{U}(2n,\mathbb{R}) \cap \operatorname{Sp}(2n,\mathbb{R}) = \mathrm{O}(2n) \cap \operatorname{Sp}(2n,\mathbb{R}), </math>such that<math display="block">S = U^\text{T} D U \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}),</math>where the diagonal elements of <math>D</math> are the [[Eigenvalues and eigenvectors|eigenvalues]] of <math>S</math>.<ref name=":0">{{Cite book|title=Symplectic Methods in Harmonic Analysis and in Mathematical Physics - Springer|last=de Gosson|first=Maurice A.|language=en|doi=10.1007/978-3-7643-9992-4|year = 2011|isbn = 978-3-7643-9991-7}}</ref><ref>{{cite arXiv|first1=Martin|last1=Houde|first2=Will|last2=McCutcheon|first3=Nicolas|last3=Quesada|title=Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson|at= Sec. V, p. 5 |date=13 March 2024|eprint=2403.04596}}</ref>

|Any real symplectic matrix {{math|S}} has a [[polar decomposition]] of the form:<ref name=":0" /><math display="block">S = UR,</math>where<math display="block">U \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{U}(2n,\mathbb{R}),</math> and<math display="block">R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}).</math>

|Any real symplectic matrix can be decomposed as a product of three matrices:<math display="block">S = O\begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}O',</math>where <math>O</math> and <math>O'</math> are both symplectic and [[Orthogonal matrix|orthogonal]], and <math>D</math> is [[Positive-definite matrix|positive-definite]] and [[Diagonal matrix|diagonal]].<ref>{{cite arXiv|first1=Alessandro|last1=Ferraro|first2=Stefano|last2=Olivares|first3=Matteo G. A.|last3=Paris|title=Gaussian states in continuous variable quantum information|at= Sec. 1.3, p. 4 |date=31 March 2005|eprint=quant-ph/0503237}}</ref> This decomposition is closely related to the [[singular value decomposition]] of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.

| The set of orthogonal symplectic matrices forms a (maximal) compact subgroup of the symplectic group.<ref name=":4">{{ cite book|title= Quantum Continuous Variables |last=Serafini|first=Alessio|language=en|doi=10.1201/9781003250975|year = 2023|isbn = 9781003250975}}</ref> This set is isomorphic to the set of unitary matrices of dimension <math> n </math>, <math>\mathrm{U}(2n,\mathbb{R}) \cap \operatorname{Sp}(2n,\mathbb{R}) = \mathrm{O}(2n) \cap \operatorname{Sp}(2n,\mathbb{R}) \cong \mathrm{U}(n,\mathbb{C})</math>. Every symplectic orthogonal matrix can be written as
{{NumBlk||<math display="block">
\begin{pmatrix}
\Re(V) & -\Im(V) \\
\Im(V) & \Re(V)
\end{pmatrix} = \left[\frac{1}{\sqrt{2}}\begin{pmatrix}
I_n & i  I_n \\
I_n & -i I_n
\end{pmatrix}  \right]^\dagger \begin{pmatrix}
V & 0 \\
0 & V^*
\end{pmatrix} \left[\frac{1}{\sqrt{2}}\begin{pmatrix}
I_n & i  I_n \\
I_n & -i I_n
\end{pmatrix}  \right],
</math><br/>|{{EquationRef|2}}}}
with <math> V \in \mathrm{U}(n,\mathbb{C})</math>.
This equation implies that every symplectic orthogonal matrix has determinant equal to +1 and thus that this is true for all symplectic matrices as its polar decomposition is itself given in terms symplectic matrices.
}}