Symplectic matrix

From Wikipedia, the free encyclopedia

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 M^T 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 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.

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

$A^{T}D-C^{T}B=I$

$A^{T}C=C^{T}A$

$D^{T}B=B^{T}D.$

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

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

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, represented by a matrix A, we have

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

$M\mapsto A^{{-1}}MA.$

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.

Complex matrices

If instead M is a 2n×2n matrix with complex entries, the definition is not standard throughout the literature. Many authors ^[1] 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 ^[2] retain the definition (1) for complex matrices and call matrices satisfying (2) conjugate symplectic.

References

↑ 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

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.