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 for any field. 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.

When the underlying field is real or complex, elementary proof is obtained by factoring the inequality $\det(M^{\text{T}}M+I)\geq 1$ .^[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 $U(2 n, R)$ such that

S=U^{{\text{T}}}DU\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

.^[2]

Any real symplectic matrix $S$ has a polar decomposition^[3] 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.^[4] This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.

Complex matrices

If instead M is a 2n×2n matrix with complex entries, the definition is not standard throughout the literature. Many authors ^[5] 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 ^[6] retain the definition (1) for complex matrices and call matrices satisfying (2) conjugate symplectic.

References

↑ Rim, D. (2015). "An Elementary Proof That Symplectic Matrices Have Determinant One". arXiv:1505.04240.
↑ [webzoom.freewebs.com/cvdegosson/symplectic%20group.pdf "Symplectic Group"], Retrieved on 30 January 2015.
↑ [webzoom.freewebs.com/cvdegosson/symplectic%20group.pdf "Symplectic Group"], Retrieved on 30 January 2015.
↑ Ferraro et. al. 2005 Section 1.3.
↑ Xu, H. G. (July 15, 2003). "An SVD-like matrix decomposition and its applications". Linear Algebra and its Applications. 368: 1–24. doi:10.1016/S0024-3795(03)00370-7.
↑ Mackey, D. S.; Mackey, N. (2003). "On the Determinant of Symplectic Matrices". Numerical Analysis Report. 422. Manchester, England: Manchester Centre for Computational Mathematics.

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