1. Properties

2. Symplectic transformations

3. The matrix Ω

4. Diagonalisation and decomposition

5. Complex matrices

6. Applications

7. See also

8. References

9. External links

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

where M^T denotes the transpose of M and Ω is a fixed {{math|2n × 2n}} nonsingular, skew-symmetric matrix. This definition can be extended to {{math|2n × 2n}} matrices with entries in other fields, such as the complex numbers.

Typically Ω is chosen to be the block matrix

where I_n is the {{math|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 determinant 1, and the {{math|2n × 2n}} symplectic matrices with real entries form a subgroup of the special linear group SL(2n, R) under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension {{nowrap|n(2n + 1)}}, and is denoted 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.

Examples of symplectic matrices include the identity matrix and the matrix .

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.

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

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

When the underlying field is real or complex, one can also show this by factoring the inequality .^[1]

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

where A, B, C, D are n×n matrices. The condition for M to be symplectic is equivalent to the two following equivalent conditions^[2]

symmetric, and

symmetric, and

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

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

this gives equations of the form

where 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.

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.

Diagonalisation and decomposition

• For any positive definite real symplectic matrix {{math|S}} there exists {{math|U}} in {{math|U(2n,R)}} such that

where the diagonal elements of {{math|D}} are the eigenvalues of {{math|S}}.^[3]

• Any real symplectic matrix {{math|S}} has a polar decomposition of the form:^[3]

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

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

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 ({{EquationNote|1}}) for complex matrices and call matrices satisfying ({{EquationNote|3}}) conjugate symplectic.

Applications

Transformations described by symplectic matrices play an important role in quantum optics and in continuous-variable quantum information theory. For instance, symplectic matrices can be used to describe Gaussian (Bogoliubov) transformations of a quantum state of light.^[7] In turn, the Bloch-Messiah decomposition ({{EquationNote|2}}) means that such an arbitrary Gaussian transformation can be represented as a set of two passive linear-optical interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix D).^[8] In fact, one can circumvent the need for such in-line active squeezing transformations if two-mode squeezed vacuum states are available as a prior resource only.^[9]

See also

• symplectic vector space
• symplectic group
• symplectic representation
• orthogonal matrix
• unitary matrix
• Hamiltonian mechanics
• Linear complex structure

References

External links

• {{planetmath reference|id=4140|title=Symplectic matrix}}
• {{planetmath reference|id=7455|title=The characteristic polynomial of a symplectic matrix is a reciprocal polynomial}}

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