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# Symplectic matrix

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

\( M^T \Omega M = \Omega\,. \) (1)

where MT denotes the transpose of M and Ω is a fixed 2n×2n nonsingular, skew-symmetric 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 2*n*×2*n* symplectic matrices with real entries form a subgroup of the special linear group SL(2*n*, *R*) under matrix multiplication, specifically a connected noncompact real Lie group of real dimension *n*(2*n* + 1), the symplectic group Sp(2*n*, **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 2x2-matrices consisting in the identity matrix, the upper triagonal matrix and the lower triangular matrix, each with entries 0 and 1.

Properties

Every symplectic matrix is invertible with the inverse matrix given by

\( M^{-1} = \Omega^{-1} M^T \Omega.\)

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. The symplectic group has dimension n(2n + 1).

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

\( \mbox{Pf}(M^T \Omega M) = \det(M)\mbox{Pf}(\Omega).\)

Since \( M^T \Omega M = \Omega and \mbox{Pf}(\Omega) \neq 0 \) 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

\( A^TD - C^TB = I\)

\( A^TC = C^TA\)

\( D^TB = B^TD.\)

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

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

Symplectic transformations

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 is a 2n-dimensional vector space V equipped with a nondegenerate, skew-symmetric 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^T \Omega M = \Omega.

Under a change of basis, represente\)d by a matrix A, we have

\( \Omega \mapsto A^T \Omega A\)

\( M \mapsto A^{-1} M A.\)

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, skew-symmetric matrix Ω. As explained in the previous section, Ω 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 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 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 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 skew-symmetric bilinear form. One could easily choose bases in which J is not skew-symmetric 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

\( S = U^TDU \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}),\)

where the diagonal elements of D are the eigenvalues of S.[1]

Any real symplectic matrix S has a polar decomposition[2] of the form:

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

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 positive-definite and diagonal.[3] This decomposition is closely related to the singular value decomposition of a matrix. It is known as an 'Euler' or 'Bloch-Messiah' 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

Portal icon Mathematics portal

symplectic vector space

symplectic group

symplectic representation

orthogonal matrix

unitary matrix

Hamiltonian mechanics

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.

Ferraro et. al. 2005 Section 1.3.

Xu, H. G. (July 15, 2003). "An SVD-like matrix decomposition and its applications". Linear Algebra and its Applications 368: 1–24. doi:10.1016/S0024-3795(03)00370-7.

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

Symplectic matrix at PlanetMath.org.

The characteristic polynomial of a symplectic matrix is a reciprocal polynomial at PlanetMath.org.

See also: symplectic category

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