In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition
where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.
Typically is chosen to be the block matrix
Every symplectic matrix has determinant , and the symplectic matrices with real entries form a subgroup of the general linear group 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 , and is denoted . 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
Every symplectic matrix is invertible with the inverse matrix given by
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
When the underlying field is real or complex, one can also show this by factoring the inequality .
Suppose Ω is given in the standard form and let be a block matrix given by
where are matrices. The condition for to be symplectic is equivalent to the two following equivalent conditions
When these conditions reduce to the single condition . Thus a matrix is symplectic iff it has unit determinant.
With in standard form, the inverse of is given by
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 -dimensional vector space equipped with a nondegenerate, skew-symmetric bilinear form called the symplectic form.
A symplectic transformation is then a linear transformation which preserves , i.e.
Fixing a basis for , can be written as a matrix and as a matrix . The condition that 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 to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.
Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix . As explained in the previous section, 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 given above is the block diagonal form
This choice differs from the previous one by a permutation of basis vectors.
Sometimes the notation is used instead of 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 but represents a very different structure. A complex structure is the coordinate representation of a linear transformation that squares to , whereas is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which is not skew-symmetric or does not square to .
Given a hermitian structure on a vector space, and are related via
where is the metric. That and 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.
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.
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 (3) conjugate symplectic.
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 (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). 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.
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