In mathematics, a **symplectic matrix** is a matrix with real entries that satisfies the condition

(1) |

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 where is the identity matrix. The matrix has determinant and its inverse is .

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
where is the set of symmetric matrices. Then, is generated by the set^{[1]}^{p. 2}
of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in and together, along with some power of .

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 .

When the underlying field is real or complex, one can also show this by factoring the inequality .^{[2]}

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^{[3]}

symmetric, and

symmetric, and

The second condition comes from the fact that if is symplectic, then is also symplectic. 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 The group has dimension . This can be seen by noting that is anti-symmetric. Since the space of anti-symmetric matrices has dimension the identity imposes constraints on the coefficients of and leaves with independent coefficients.

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.

- For any positive definite symmetric real symplectic matrix S there exists
*U*in such that

where the diagonal elements of D are the eigenvalues of S.^{[4]}

- Any real symplectic matrix S has a polar decomposition of the form:
^{[4]}

- for and

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

(2) |

such that O and O' are both symplectic and orthogonal and D is positive-definite and diagonal.^{[5]} 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 2*n* × 2*n* matrix with complex entries, the definition is not standard throughout the literature. Many authors ^{[6]} adjust the definition above to

(3) |

where *M ^{*}* denotes the conjugate transpose of

Other authors ^{[7]} 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.^{[8]} 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*).^{[9]} 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.^{[10]}