Symplectic matrix

In mathematics, a symplectic matrix is a formula_1 matrix formula_2 with real entries that satisfies the condition
where formula_3 denotes the transpose of formula_2 and formula_5 is a fixed formula_1 nonsingular, skew-symmetric matrix. This definition can be extended to formula_1 matrices with entries in other fields, such as the complex numbers, finite fields, "p"-adic numbers, and function fields.
Typically formula_5 is chosen to be the block matrix
formula_9
where formula_10 is the formula_11 identity matrix. The matrix formula_5 has determinant formula_13 and its inverse is formula_14.
Properties.
Generators for symplectic matrices.
Every symplectic matrix has determinant formula_13, and the formula_1 symplectic matrices with real entries form a subgroup of the general linear group formula_17 under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension formula_18, and is denoted formula_19. The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.
This symplectic group has a distinguished set of generators, which can be used to find all possible symplectic matrices. This includes the following sets
formula_20
where formula_21 is the set of formula_11 symmetric matrices. Then, formula_19 is generated by the setp. 2
formula_24
of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in formula_25 and formula_26 together, along with some power of formula_5.
Inverse matrix.
Every symplectic matrix is invertible with the inverse matrix given by
formula_28
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.
Determinantal properties.
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
formula_29
Since formula_30 and formula_31 we have that formula_32.
When the underlying field is real or complex, one can also show this by factoring the inequality formula_33.
Block form of symplectic matrices.
Suppose Ω is given in the standard form and let formula_2 be a formula_1 block matrix given by
formula_36
where formula_37 are formula_11 matrices. The condition for formula_2 to be symplectic is equivalent to the two following equivalent conditionsformula_40 symmetric, and formula_41formula_42 symmetric, and formula_43When formula_44 these conditions reduce to the single condition formula_32. Thus a formula_46 matrix is symplectic iff it has unit determinant.
Inverse matrix of block matrix.
With formula_5 in standard form, the inverse of formula_2 is given by
formula_49
The group has dimension formula_18. This can be seen by noting that formula_51 is anti-symmetric. Since the space of anti-symmetric matrices has dimension formula_52 the identity formula_53 imposes formula_54 constraints on the formula_55 coefficients of formula_2 and leaves formula_2 with formula_18 independent coefficients.
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 formula_59 is a formula_60-dimensional vector space formula_61 equipped with a nondegenerate, skew-symmetric bilinear form formula_62 called the symplectic form.
A symplectic transformation is then a linear transformation formula_63 which preserves formula_62, i.e.
Fixing a basis for formula_61, formula_62 can be written as a matrix formula_5 and formula_69 as a matrix formula_2. The condition that formula_69 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 formula_5 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 formula_5. As explained in the previous section, formula_5 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 formula_5 given above is the block diagonal form
This choice differs from the previous one by a permutation of basis vectors.
Sometimes the notation formula_80 is used instead of formula_5 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 formula_5 but represents a very different structure. A complex structure formula_80 is the coordinate representation of a linear transformation that squares to formula_84, whereas formula_5 is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which formula_80 is not skew-symmetric or formula_5 does not square to formula_84.
Given a hermitian structure on a vector space, formula_80 and formula_5 are related via
where formula_92 is the metric. That formula_80 and formula_5 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.
 such that and are both symplectic and orthogonal and 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 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 () for complex matrices and call matrices satisfying () "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. In turn, the Bloch-Messiah decomposition () 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"). 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.