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
- <math>M^T \Omega M = \Omega.<math>
where MT denotes the transpose of M and Ω is the 2n×2n skew-symmetric matrix
- <math>\Omega =
\begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}<math> Here In is the n×n identity matrix. Note that Ω has determinant +1 and squares to minus the identity: Ω2 = −I2n.
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
- <math>\Omega = \begin{bmatrix}
\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\
& \ddots & \\
0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{bmatrix}<math> 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.
Properties
Every symplectic matrix has an inverse which is given by
- <math>M^{-1} = \Omega^{-1} M^T \Omega<math>
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
- <math>\mbox{Pf}(M^T \Omega M) = \det(M)\mbox{Pf}(\Omega).<math>
Since <math>M^T \Omega M = \Omega<math> and <math>\mbox{Pf}(\Omega) \neq 0<math> we have that det(M) = 1.
Let M be a 2n×2n block matrix given by
- <math>M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}<math>
where A, B, C, D are n×n matrices. Then the condition for M to be symplectic is equivalent to the conditions
- <math>A^TD - C^TB = 1<math>
- <math>A^TC = C^TA<math>
- <math>D^TB = B^TD.<math>
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 f : V → V which preserves ω, i.e.
- <math>\omega(f(x), f(y)) = \omega(x, y).<math>
Fixing a basis for V, ω can be written as a matrix Ω and f as a matrix M. The condition that f be a symplectic transformation is precisely the condition that M be a symplectic matrix:
- <math>M^T \Omega M = \Omega.<math>
Under a change of basis, represented by a matrix A, we have
- <math>\Omega \mapsto A^T \Omega A<math>
- <math>M \mapsto A^{-1} M A.<math>
One can always bring Ω to either of the standard forms given in the introduction by a suitable choice of A.