# Symplectic matrix

In mathematics, a **symplectic matrix** is a 2*n*×2*n* matrix *M* with real entries that satisfies the condition
Template:NumBlk
where *M ^{T}* denotes the transpose of

*M*and Ω is a fixed 2

*n*×2

*n*nonsingular, skew-symmetric matrix. This definition can be extended to 2

*n*×2

*n*matrices with entries in other fields, e.g. the complex numbers.

Typically Ω is chosen to be the block matrix

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 2*n*×2*n* symplectic matrices with real entries form a subgroup of the special linear group SL(2*n*, *R*) under matrix multiplication, specifically a connected noncompact real Lie group of real dimension *n*(2*n* + 1), the symplectic group Sp(2*n*, **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

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*(2*n* + 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

Since and we have that det(*M*) = 1.

Suppose Ω is given in the standard form and let *M* be a 2*n*×2*n* block matrix given by

where *A, B, C, D* are *n*×*n* matrices. The condition for *M* to be symplectic is equivalent to the conditions

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

## 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 2*n*-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.

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:

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

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

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

where 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 ^{[1]} adjust the definition above to
Template:NumBlk
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 ^{[2]} retain the definition (Template:EquationNote) for complex matrices and call matrices satisfying (Template:EquationNote) *conjugate symplectic*.

## See also

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