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
In other words, any symplectic matrix can be constructed by multiplying matrices in
This gives the set of all symplectic matrices the structure of a group.
It follows easily from the definition that the determinant of any symplectic matrix is ±1.
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
matrix is symplectic iff it has unit determinant.
Since the space of anti-symmetric matrices has dimension
In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finite-dimensional vector spaces.
equipped with a nondegenerate, skew-symmetric bilinear form
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
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.
given above is the block diagonal form This choice differs from the previous one by a permutation of basis vectors.
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
is the coordinate representation of a linear transformation that squares to
is the coordinate representation of a nondegenerate skew-symmetric bilinear form.
Given a hermitian structure on a vector space,
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.
If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature.
Many authors [8] 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 [9] 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.
[10] 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).
[11] 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.