In mathematics, a symplectic matrix is a 2n×2n matrix M with real entries that satisfies the condition

M^\text{T} \Omega M = \Omega\,,


(1)

where M^{T} denotes the transpose of M and Ω is a fixed 2n×2n nonsingular, skewsymmetric matrix. This definition can be extended to 2n×2n matrices with entries in other fields, e.g. the complex numbers.
Typically Ω is chosen to be the block matrix

\Omega = \begin{bmatrix} 0 & I_n \\ I_n & 0 \\ \end{bmatrix}
where I_{n} is the n×n identity matrix. The matrix Ω has determinant +1 and has an inverse given by Ω^{−1} = Ω^{T} = −Ω.
Every symplectic matrix has unit determinant, and the 2n×2n symplectic matrices with real entries form a subgroup of the special linear group SL(2n, R) under matrix multiplication, specifically a connected noncompact real Lie group of real dimension n(2n + 1), the symplectic group Sp(2n, R). The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.
An example of a group of symplectic matrices is the group of three symplectic 2x2matrices consisting in the identity matrix, the upper triagonal matrix and the lower triangular matrix, each with entries 0 and 1.
Contents

Properties 1

Symplectic transformations 2

The matrix Ω 3

Diagonalisation and decomposition 4

Complex matrices 5

See also 6

References 7

External links 8
Properties
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. One way to see this is through the use of the Pfaffian and the identity

\mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).
Since M^\text{T} \Omega M = \Omega and \mbox{Pf}(\Omega) \neq 0 we have that det(M) = 1.
Suppose Ω is given in the standard form and let M be a 2n×2n block matrix given by

M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}
where A, B, C, D are n×n matrices. The condition for M to be symplectic is equivalent to the conditions

A^\text{T} D  C^\text{T} B = I

A^\text{T} C = C^\text{T} A

D^\text{T} B = B^\text{T} D.
When n = 1 these conditions reduce to the single condition det(M) = 1. Thus a 2×2 matrix is symplectic iff it has unit determinant.
With Ω 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 the group condition implies that

\Omega M^\text{T} \Omega M = I
this gives equations of the form

\delta_{ij} = \sum_{k=1}^n m_{k,i+n}m_{n+k,j}  m_{n+k,i+n}m_{n,j}  m_{k,i}m_{n+k,j} + m_{k,i}m_{k,j}
where m_{ij} is the i,jth element of M. The sum is antisymmetric with respect to indices i,j, and since the left hand side is zero when i differs from j, this leaves n(2n1) independent equations.
Symplectic transformations
In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finitedimensional vector spaces. The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space is a 2ndimensional vector space V equipped with a nondegenerate, skewsymmetric bilinear form ω called the symplectic form.
A symplectic transformation is then a linear transformation L : V → V which preserves ω, i.e.

\omega(Lu, Lv) = \omega(u, v).
Fixing a basis for V, ω can be written as a matrix Ω 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^\text{T} \Omega M = \Omega.
Under a change of basis, represented by a matrix A, we have

\Omega \mapsto A^\text{T} \Omega A

M \mapsto A^{1} M A.
One can always bring Ω to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.
The matrix Ω
Symplectic matrices are defined relative to a fixed nonsingular, skewsymmetric matrix Ω. As explained in the previous section, Ω can be thought of as the coordinate representation of a nondegenerate skewsymmetric 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 Ω given above is the block diagonal form

\Omega = \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 Ω for the skewsymmetric 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 Ω but represents a very different structure. A complex structure J is the coordinate representation of a linear transformation that squares to −1, whereas Ω is the coordinate representation of a nondegenerate skewsymmetric bilinear form. One could easily choose bases in which J is not skewsymmetric or Ω does not square to −1.
Given a hermitian structure on a vector space, J and Ω are related via

\Omega_{ab} = g_{ac}{J^c}_b
where g_{ac} is the metric. That J and Ω 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.
Diagonalisation and decomposition

For any positive definite real symplectic matrix S there exists U in U(2n,R) such that

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}),
where the diagonal elements of D are the eigenvalues of S.^{[1]}

S=UR \quad \text{for} \quad U \in \operatorname{U}(2n,\mathbb{R}) \text{ and } R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}).

Any real symplectic matrix can be decomposed as a product of three matrices:

S = O\begin{pmatrix}D & 0 \\ 0 & D^{1}\end{pmatrix}O',
such that O and O' are both symplectic and orthogonal and D is positivedefinite and diagonal.^{[3]} This decomposition is closely related to the singular value decomposition of a matrix. It is known as an 'Euler' or 'BlochMessiah' decomposition and has an intuitive link with the Euler decomposition of a rotation.
Complex matrices
If instead M is a 2n×2n matrix with complex entries, the definition is not standard throughout the literature. Many authors ^{[4]} adjust the definition above to

M^* \Omega M = \Omega\,.


(2)

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 ^{[5]} retain the definition (1) for complex matrices and call matrices satisfying (2) conjugate symplectic.
See also
References

^ [webzoom.freewebs.com/cvdegosson/symplectic%20group.pdf "Symplectic Group"], Retrieved on 30 January 2015.

^ [webzoom.freewebs.com/cvdegosson/symplectic%20group.pdf "Symplectic Group"], Retrieved on 30 January 2015.

^ 2005et. al.Ferraro Section 1.3.

^ Xu, H. G. (July 15, 2003). "An SVDlike matrix decomposition and its applications". Linear Algebra and its Applications 368: 1–24.

^ Mackey, D. S.; Mackey, N. (2003). "On the Determinant of Symplectic Matrices". Numerical Analysis Report 422. Manchester, England: Manchester Centre for Computational Mathematics.
External links
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.