Symplectic matrix

Let M be a 2n×2n matrix with real entries. Then M is called a symplectic matrix if it satisfies the condition

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

(1)

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

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.

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

$\Omega \mapsto A^T \Omega A$

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

One can always bring Ω to either of the standard forms given in the introduction 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 adjust the definition above to

$M^* \Omega M = \Omega\,.$

(2)

where M^* denotes the conjugate transpose of M. Other authors retain the definition (1) for complex matrices and call matrices satisfying (2) conjugate symplectic.

See also

• symplectic vector space
• symplectic group
• symplectic representation
• orthogonal matrix
• unitary matrix
• Hamiltonian mechanics

External links

• Symplectic matrix on PlanetMath
• The characteristic polynomial of a symplectic matrix is a reciprocal polynomial on PlanetMath