symplectic matrix

Every symplectic matrix has determinant $+1$, and the $2n\backslash times\; 2n$ symplectic matrices with real entries form a subgroup of the general linear group $\backslash mathrm\{GL\}(2n;\backslash mathbb\{R\})$ under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension $n(2n+1)$, and is denoted $\backslash mathrm\{Sp\}(2n;\backslash mathbb\{R\})$. 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 set of generators, which can be used to find all possible symplectic matrices. This includes the following sets

$$\backslash begin\{align\}$$

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}

where $\backslash text\{Sym\}(n;\backslash mathbb\{R\})$ is the set of $n\backslash times\; n$ symmetric matrices. Then, $\backslash mathrm\{Sp\}(2n;\backslash mathbb\{R\})$ is generated by the set{{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}}^{p. 2}

$$\backslash \{\backslash Omega\; \backslash \}\; \backslash cup\; D(n)\; \backslash cup\; N(n)$$

of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in $D(n)$ and $N(n)$ together, along with some power of $\backslash Omega$.

Every symplectic matrix is invertible with the inverse matrix given by

M^{-1} = \Omega^{-1} M^\text{T} \Omega.

Furthermore, the product of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a group. There exists a natural manifold structure on this group which makes it into a (real or complex) Lie group called the symplectic group.

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

$$\backslash mbox\{Pf\}(M^\backslash text\{T\}\; \backslash Omega\; M)\; =\; \backslash det(M)\backslash mbox\{Pf\}(\backslash Omega).$$

Since $M^\backslash text\{T\}\; \backslash Omega\; M\; =\; \backslash Omega$ and $\backslash mbox\{Pf\}(\backslash Omega)\; \backslash neq\; 0$ we have that $\backslash det(M)\; =\; 1$.

When the underlying field is real or complex, one can also show this by factoring the inequality $\backslash det(M^\backslash text\{T\}\; M\; +\; I)\; \backslash ge\; 1$.{{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 }}

Suppose Ω is given in the standard form and let $M$ be a $2n\backslash times\; 2n$ block matrix given by

where $A,B,C,D$ are $n\backslash times\; n$ matrices. The condition for $M$ to be symplectic is equivalent to the two following equivalent conditions{{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}}

$A^\backslash text\{T\}C,B^\backslash text\{T\}D$ symmetric, and $A^\backslash text\{T\}\; D\; -\; C^\backslash text\{T\}\; B\; =\; I$

$AB^\backslash text\{T\},CD^\backslash text\{T\}$ symmetric, and $AD^\backslash text\{T\}\; -\; BC^\backslash text\{T\}\; =\; I$

With $\backslash Omega$ in standard form, the inverse of $M$ is given by

M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}.

The group has dimension $n(2n+1)$. This can be seen by noting that $(\; M^\backslash text\{T\}\; \backslash Omega\; M)^\backslash text\{T\}\; =\; -M^\backslash text\{T\}\; \backslash Omega\; M$ is anti-symmetric. Since the space of anti-symmetric matrices has dimension $\backslash binom\{2n\}\{2\},$ the identity $M^\backslash text\{T\}\; \backslash Omega\; M\; =\; \backslash Omega$ imposes $2n\; \backslash choose\; 2$ constraints on the $(2n)^2$ coefficients of $M$ and leaves $M$ with $n(2n+1)$ independent coefficients.

In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finite-dimensional vector spaces. The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space $(V,\backslash omega)$ is a $2n$-dimensional vector space $V$ equipped with a nondegenerate, skew-symmetric bilinear form $\backslash omega$ called the symplectic form.

A symplectic transformation is then a linear transformation $L:V\backslash to\; V$ which preserves $\backslash omega$, i.e.

$$\backslash omega(Lu,\; Lv)\; =\; \backslash omega(u,\; v).$$

Fixing a basis for $V$, $\backslash omega$ can be written as a matrix $\backslash Omega$ and $L$ as a matrix $M$. The condition that $L$ be a symplectic transformation is precisely the condition that M be a symplectic matrix:

$$M^\backslash text\{T\}\; \backslash Omega\; M\; =\; \backslash Omega.$$

Under a change of basis, represented by a matrix A, we have

$$\backslash Omega\; \backslash mapsto\; A^\backslash text\{T\}\; \backslash Omega\; A$$

$$M\; \backslash mapsto\; A^\{-1\}\; M\; A.$$

One can always bring $\backslash Omega$ to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.

Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix $\backslash Omega$. As explained in the previous section, $\backslash Omega$ can be thought of as the coordinate representation of a nondegenerate skew-symmetric bilinear form. It is a basic result in linear algebra that any two such matrices differ from each other by a change of basis.

The most common alternative to the standard $\backslash Omega$ given above is the block diagonal form

$$\backslash Omega\; =\; \backslash begin\{bmatrix\}$$

\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\

& \ddots & \\

0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix}

\end{bmatrix}.

This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation $J$ is used instead of $\backslash Omega$ for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a complex structure, which often has the same coordinate expression as $\backslash Omega$ but represents a very different structure. A complex structure $J$ is the coordinate representation of a linear transformation that squares to $-I\_n$, whereas $\backslash Omega$ is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which $J$ is not skew-symmetric or $\backslash Omega$ does not square to $-I\_n$.

Given a hermitian structure on a vector space, $J$ and $\backslash Omega$ are related via

$$\backslash Omega\_\{ab\}\; =\; -g\_\{ac\}\{J^c\}\_b$$

where $g\_\{ac\}$ is the metric. That $J$ and $\backslash Omega$ usually have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric g is usually the identity matrix.

{{bullet list

|For any positive definite symmetric $2n\backslash times\; 2n$ real symplectic matrix $S$, there is a symplectic unitary $U$, $$U\; \backslash in\; \backslash mathrm\{U\}(2n,\backslash mathbb\{R\})\; \backslash cap\; \backslash operatorname\{Sp\}(2n,\backslash mathbb\{R\})\; =\; \backslash mathrm\{O\}(2n)\; \backslash cap\; \backslash operatorname\{Sp\}(2n,\backslash mathbb\{R\}),$$such that$$S\; =\; U^\backslash text\{T\}\; D\; U\; \backslash quad\; \backslash text\{for\}\; \backslash quad\; D\; =\; \backslash operatorname\{diag\}(\backslash lambda\_1,\backslash ldots,\backslash lambda\_n,\backslash lambda\_1^\{-1\},\backslash ldots,\backslash lambda\_n^\{-1\}),$$where the diagonal elements of $D$ are the eigenvalues of $S$.{{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}}{{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}}

|Any real symplectic matrix {{math|S}} has a polar decomposition of the form:$$S\; =\; UR,$$where$$U\; \backslash in\; \backslash operatorname\{Sp\}(2n,\backslash mathbb\{R\})\backslash cap\backslash operatorname\{U\}(2n,\backslash mathbb\{R\}),$$ and$$R\; \backslash in\; \backslash operatorname\{Sp\}(2n,\backslash mathbb\{R\})\backslash cap\backslash operatorname\{Sym\}\_+(2n,\backslash mathbb\{R\}).$$

|Any real symplectic matrix can be decomposed as a product of three matrices:where $O$ and $O\text{'}$ are both symplectic and orthogonal, and $D$ is positive-definite and diagonal.{{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}} 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.{{ cite book|title= Quantum Continuous Variables |last=Serafini|first=Alessio|language=en|doi=10.1201/9781003250975|year = 2023|isbn = 9781003250975}} This set is isomorphic to the set of unitary matrices of dimension $n$, $\backslash mathrm\{U\}(2n,\backslash mathbb\{R\})\; \backslash cap\; \backslash operatorname\{Sp\}(2n,\backslash mathbb\{R\})\; =\; \backslash mathrm\{O\}(2n)\; \backslash cap\; \backslash operatorname\{Sp\}(2n,\backslash mathbb\{R\})\; \backslash cong\; \backslash mathrm\{U\}(n,\backslash mathbb\{C\})$. Every symplectic orthogonal matrix can be written as

{{NumBlk||$$$$

\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],

|{{EquationRef|2}}}}

with $V\; \backslash in\; \backslash mathrm\{U\}(n,\backslash mathbb\{C\})$.

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.

}}

If instead M is a {{nowrap|2n × 2n}} matrix with complex entries, the definition is not standard throughout the literature. Many authors {{cite journal|last = Xu|first= H. G.|title= An SVD-like matrix decomposition and its applications|journal= Linear Algebra and Its Applications|date= July 15, 2003|volume= 368|pages=1–24|doi = 10.1016/S0024-3795(03)00370-7|hdl= 1808/374|hdl-access= free}} adjust the definition above to

{{NumBlk||$$M^*\; \backslash Omega\; M\; =\; \backslash Omega\backslash ,.$$|{{EquationRef|3}}}}

where M^* denotes the conjugate transpose of M. In this case, the determinant may not be 1, but will have absolute value 1. In the 2×2 case (n=1), M will be the product of a real symplectic matrix and a complex number of absolute value 1.

Other authors {{Cite report|last1=Mackey |last2= Mackey|first1= D. S. |first2= N.|title= On the Determinant of Symplectic Matrices|year= 2003

|type=Numerical Analysis Report 422|publisher=Manchester Centre for Computational Mathematics|location=Manchester, England

}} retain the definition ({{EquationNote|1}}) for complex matrices and call matrices satisfying ({{EquationNote|3}}) conjugate symplectic.

Transformations described by symplectic matrices play an important role in quantum optics and in continuous-variable quantum information theory. For instance, symplectic matrices can be used to describe Gaussian (Bogoliubov) transformations of a quantum state of light.{{Cite journal|last1=Weedbrook|first1=Christian|last2=Pirandola|first2=Stefano|last3=García-Patrón|first3=Raúl|last4=Cerf|first4=Nicolas J.|last5=Ralph|first5=Timothy C.|last6=Shapiro|first6=Jeffrey H.|last7=Lloyd|first7=Seth|date=2012|title=Gaussian quantum information|journal=Reviews of Modern Physics|volume=84|issue=2|pages=621–669|arxiv=1110.3234|doi=10.1103/RevModPhys.84.621|bibcode=2012RvMP...84..621W|s2cid=119250535 }} In turn, the Bloch-Messiah decomposition ({{EquationNote|2}}) means that such an arbitrary Gaussian transformation can be represented as a set of two passive linear-optical interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix D).{{Cite journal|last=Braunstein|first=Samuel L.|title=Squeezing as an irreducible resource|date=2005|journal=Physical Review A|volume=71|issue=5|pages=055801|doi=10.1103/PhysRevA.71.055801|arxiv=quant-ph/9904002|bibcode=2005PhRvA..71e5801B|s2cid=16714223 }} In fact, one can circumvent the need for such in-line active squeezing transformations if two-mode squeezed vacuum states are available as a prior resource only.{{cite journal|last1=Chakhmakhchyan|first1=Levon|last2=Cerf|first2=Nicolas|title= Simulating arbitrary Gaussian circuits with linear optics|journal=Physical Review A|date=2018|volume=98|issue=6|page=062314|doi=10.1103/PhysRevA.98.062314|arxiv=1803.11534|bibcode=2018PhRvA..98f2314C|s2cid=119227039 }}

{{Portal|Mathematics}}

• Symplectic vector space
• Symplectic group
• Symplectic representation
• Orthogonal matrix
• Unitary matrix
• Hamiltonian mechanics
• Linear complex structure
• Williamson theorem

{{reflist}}

{{Matrix classes}}

Category:Matrices (mathematics)

Category:Symplectic geometry