Symplectic matrix

Properties

Every symplectic matrix is invertible with the inverse matrix given by

M − 1 = Ω − 1 M T Ω .

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

Pf ( M T Ω M ) = det ( M ) Pf ( Ω ) .

Since M T Ω M = Ω and Pf ( Ω ) ≠ 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 T M + I ) ≥ 1 .

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

M = ( A B C D )

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

A T D − C T B = I A T C = C T A D T B = B 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 = Ω − 1 M T Ω = ( D T − B T − C T A T ) .

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

Ω M T Ω M = − I

this gives equations of the form

− δ i j = ∑ 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 i j 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.

ω ( L u , L v ) = ω ( 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 Ω M = Ω .

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

Ω ↦ A T Ω A M ↦ 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

Ω = [ 0 1 − 1 0 0 ⋱ 0 0 1 − 1 0 ] .

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

Ω a b = − g a c J c b

where g a c 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 T D U for D = diag ⁡ ( λ 1 , … , λ n , λ 1 − 1 , … , λ 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 = U R for U ∈ U ⁡ ( 2 n , R )  and  R ∈ Sp ⁡ ( 2 n , R ) ∩ Sym + ⁡ ( 2 n , R ) .

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

S = O ( D 0 0 D − 1 ) 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 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 adjust the definition above to

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.