Symplectic matrix

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In mathematics, a symplectic matrix is a 2n×2n matrix M (whose entries are typically either real or complex) satisfying the condition

M^T \Omega M = \Omega\,.

where MT denotes the transpose of M and Ω is a fixed nonsingular, skew-symmetric matrix. Typically Ω is chosen to be the block matrix

\Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}

where In is the n×n identity matrix. Note that Ω has determinant +1 and has an inverse given by Ω−1 = −Ω.

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[edit] Properties

Every symplectic matrix is invertible with the inverse matrix given by

M − 1 = Ω − 1MTΩ

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

Pf(MTΩM) = det(M)Pf(Ω).

Since MTΩM = Ω 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

ATD − CTB = 1
ATC = CTA
DTB = BTD.

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.

[edit] 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.

A symplectic transformation is then a linear transformation L : V → V which preserves ω, i.e.

ω(Lu,Lv) = ω(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:

MTΩM = Ω.

Under a change of basis, represented 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 of the standard forms given in the introduction by a suitable choice of A.

[edit] 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 gac 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.

[edit] See also

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