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

\( M^T \Omega M = \Omega\,. \) (1)

where MT 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

\( M^{-1} = \Omega^{-1} M^T \Omega.\)

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.

With Ω in standard form, the inverse of M is given by

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

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, represente\)d 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 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

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

Any real symplectic matrix S has a polar decomposition[2] 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.[3] 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 [4] 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 [5] retain the definition (1) for complex matrices and call matrices satisfying (2) conjugate symplectic.

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Symplectic matrix