Symplectic matrix

Encyclopedia : S : SY : SYM : Symplectic matrix

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

[\Omega =\begin0 & I_n \\-I_n & 0 \\\end]

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\begin0 & 1\\ -1 & 0\end & & 0 \\ & \ddots & \\0 & & \begin0 & 1 \\ -1 & 0\end\end]

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.

Contents

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

Properties

Every symplectic matrix has an inverse which is given by

[M^ = \Omega^ 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(M^T \Omega M) = \det(M)\mbox(\Omega).]

Since [M^T \Omega M = \Omega] and [\mbox(\Omega) \neq 0] we have that det(M) = 1.

Let M be a 2n×2n block matrix given by

[M = \beginA & B \\ C & D\end]

where A, B, C, D are n×n matrices. Then the condition for M to be symplectic is equivalent to the conditions

[A^TD - C^TB = 1]

[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.

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.

[\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^ M A.]

One can always bring Ω to either of the standard forms given in the introduction by a suitable choice of A.

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_ = g__b]

where [g_] 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.

References

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

Symplectic matrix

Properties

Symplectic transformations

Notation: J vs. Ω

See also

References