Symplectic matrix

In mathematics, a symplectic matrix is a 2n×2n matrix M with real entries that satisfies the condition

$M^\text{T} \Omega M = \Omega\,,$

(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.

An example of a group of symplectic matrices is the group of three symplectic 2x2-matrices consisting in the identity matrix, the upper triagonal matrix and the lower triangular matrix, each with entries 0 and 1.

Properties

Every symplectic matrix is invertible with the inverse matrix given by

$M^{-1} = \Omega^{-1} M^\text{T} \Omega.$

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.

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

$\mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).$

Since $M^\text{T} \Omega M = \Omega$ 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

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

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

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

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

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

The group has dimension n(2n + 1). This can be seen by noting that the group condition implies that

$\Omega M^\text{T} \Omega M = -I$

this gives equations of the form

$-\delta_{ij} = \sum_{k=1}^n m_{k,i+n}m_{n+k,j} - m_{n+k,i+n}m_{n,j} - m_{k,i}m_{n+k,j} + m_{k,i}m_{k,j}$

where $m_{ij}$ is the i,j-th element of M. The sum is antisymmetric with respect to indices i,j, and since the left hand side is zero when i differs from j, this leaves n(2n-1) independent equations.

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:

$M^\text{T} \Omega M = \Omega.$

Under a change of basis, represented by a matrix A, we have

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

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

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.

Diagonalisation and decomposition

• For any positive definite real symplectic matrix S there exists U in U(2n,R) such that

$S = U^\text{T} D U \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}),$
where the diagonal elements of D are the eigenvalues of S.

• Any real symplectic matrix S has a polar decomposition of the form:

$S=UR \quad \text{for} \quad U \in \operatorname{U}(2n,\mathbb{R}) \text{ and } R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}).$

• Any real symplectic matrix can be decomposed as a product of three matrices:

$S = O\begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}O',$
such that O and O' are both symplectic and orthogonal and D is positive-definite and diagonal. This decomposition is closely related to the singular value decomposition of a matrix. It is known as an 'Euler' or 'Bloch-Messiah' decomposition and has an intuitive link with the Euler decomposition of a rotation.

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. 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 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 at PlanetMath.org.
• The characteristic polynomial of a symplectic matrix is a reciprocal polynomial at PlanetMath.org.