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 Ω M = Ω.

where M^T denotes the transpose of M and Ω is the 2n×2n skew-symmetric matrix

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

Here I_n is the n×n identity matrix. Note that Ω has determinant +1 and squares to minus the identity: Ω² = −I_2n.

N.B. Some authors prefer to use a different Ω for the definition of symplectic matrices. The only essential property is that Ω be a nonsingular, skew-symmetric matrix. The most common alternative is the block diagonal form

$\Omega = \begin{bmatrix} \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{bmatrix}$

Note that this differs from the previous choice by a permutation of basis vectors. In fact, any choice of Ω can be brought to either of the above forms by a different choice of basis. See the abstract formulation below in the section on symplectic transformations.

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 a linear complex structure, as described below.

1 Properties
2 Symplectic transformations
3 Notation: J vs. Ω
4 See also
5 References

[edit] Properties

Every symplectic matrix is invertible with the inverse matrix given by

M - 1 = Ω - 1 M T Ω

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(M T Ω M) = det(M)Pf(Ω).

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

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. Then the condition for M to be symplectic is equivalent to the conditions

A T D - C T B = 1

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.

[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(L u, L v) = 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:

M T Ω M = 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] Notation: J vs. Ω

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 a linear complex structure, which often has the same coordinate expression but represents a very different structure. These are two different things and should be distinguished. In particular, one could easily choose a basis for which Ω² ≠ −1, whereas this is an essential quality of a complex structure. Moreover, J should be understood as a linear transformation whereas Ω is a bilinear form.

Given a hermitian structure on a vector space, J and Ω are related via

$\Omega_{ab} = g_{ac}{J^c}_b$

where $g a c$ is the metric. That J and Ω can sometimes have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric g is often the identity matrix.