# Symplectic matrix

In mathematics, a symplectic matrix is a $\displaystyle{ 2n\times 2n }$ matrix $\displaystyle{ M }$ with real entries that satisfies the condition

$\displaystyle{ M^\text{T} \Omega M = \Omega, }$

(1)

where $\displaystyle{ M^\text{T} }$ denotes the transpose of $\displaystyle{ M }$ and $\displaystyle{ \Omega }$ is a fixed $\displaystyle{ 2n\times 2n }$ nonsingular, skew-symmetric matrix. This definition can be extended to $\displaystyle{ 2n\times 2n }$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically $\displaystyle{ \Omega }$ is chosen to be the block matrix $\displaystyle{ \Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}, }$ where $\displaystyle{ I_n }$ is the $\displaystyle{ n\times n }$ identity matrix. The matrix $\displaystyle{ \Omega }$ has determinant $\displaystyle{ +1 }$ and its inverse is $\displaystyle{ \Omega^{-1} = \Omega^\text{T} = -\Omega }$.

## Properties

### Generators for symplectic matrices

Every symplectic matrix has determinant $\displaystyle{ +1 }$, and the $\displaystyle{ 2n\times 2n }$ symplectic matrices with real entries form a subgroup of the general linear group $\displaystyle{ \mathrm{GL}(2n;\mathbb{R}) }$ 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 $\displaystyle{ n(2n+1) }$, and is denoted $\displaystyle{ \mathrm{Sp}(2n;\mathbb{R}) }$. 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 \displaystyle{ \begin{align} D(n) =& \left\{ \begin{pmatrix} A & 0 \\ 0 & (A^T)^{-1} \end{pmatrix} : A \in \text{GL}(n;\mathbb{R}) \right\} \\ N(n) =& \left\{ \begin{pmatrix} I_n & B \\ 0 & I_n \end{pmatrix} : B \in \text{Sym}(n;\mathbb{R}) \right\} \end{align} } where $\displaystyle{ \text{Sym}(n;\mathbb{R}) }$ is the set of $\displaystyle{ n\times n }$ symmetric matrices. Then, $\displaystyle{ \mathrm{Sp}(2n;\mathbb{R}) }$ is generated by the set[1]p. 2 $\displaystyle{ \{\Omega \} \cup D(n) \cup N(n) }$ of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in $\displaystyle{ D(n) }$ and $\displaystyle{ N(n) }$ together, along with some power of $\displaystyle{ \Omega }$.

### Inverse matrix

Every symplectic matrix is invertible with the inverse matrix given by $\displaystyle{ M^{-1} = \Omega^{-1} M^\text{T} \Omega. }$ 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 $\displaystyle{ \mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega). }$ Since $\displaystyle{ M^\text{T} \Omega M = \Omega }$ and $\displaystyle{ \mbox{Pf}(\Omega) \neq 0 }$ we have that $\displaystyle{ \det(M)=1 }$.

When the underlying field is real or complex, one can also show this by factoring the inequality $\displaystyle{ \det(M^\text{T} M + I) \ge 1 }$.[2]

### Block form of symplectic matrices

Suppose Ω is given in the standard form and let $\displaystyle{ M }$ be a $\displaystyle{ 2n\times 2n }$ block matrix given by $\displaystyle{ M = \begin{pmatrix}A & B \\ C & D\end{pmatrix} }$

where $\displaystyle{ A,B,C,D }$ are $\displaystyle{ n\times n }$ matrices. The condition for $\displaystyle{ M }$ to be symplectic is equivalent to the two following equivalent conditions[3]

$\displaystyle{ A^\text{T}C,B^\text{T}D }$ symmetric, and $\displaystyle{ A^\text{T} D - C^\text{T} B = I }$

$\displaystyle{ AB^\text{T},CD^\text{T} }$ symmetric, and $\displaystyle{ AD^\text{T} - BC^\text{T} = I }$

When $\displaystyle{ n=1 }$ these conditions reduce to the single condition $\displaystyle{ \det(M)=1 }$. Thus a $\displaystyle{ 2\times 2 }$ matrix is symplectic iff it has unit determinant.

#### Inverse matrix of block matrix

With $\displaystyle{ \Omega }$ in standard form, the inverse of $\displaystyle{ M }$ is given by $\displaystyle{ M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}. }$ The group has dimension $\displaystyle{ n(2n+1) }$. This can be seen by noting that $\displaystyle{ ( M^\text{T} \Omega M)^\text{T} = -M^\text{T} \Omega M }$ is anti-symmetric. Since the space of anti-symmetric matrices has dimension $\displaystyle{ \binom{2n}{2}, }$ the identity $\displaystyle{ M^\text{T} \Omega M = \Omega }$ imposes $\displaystyle{ 2n \choose 2 }$ constraints on the $\displaystyle{ (2n)^2 }$ coefficients of $\displaystyle{ M }$ and leaves $\displaystyle{ M }$ with $\displaystyle{ n(2n+1) }$ 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 $\displaystyle{ (V,\omega) }$ is a $\displaystyle{ 2n }$-dimensional vector space $\displaystyle{ V }$ equipped with a nondegenerate, skew-symmetric bilinear form $\displaystyle{ \omega }$ called the symplectic form.

A symplectic transformation is then a linear transformation $\displaystyle{ L:V\to V }$ which preserves $\displaystyle{ \omega }$, i.e.

$\displaystyle{ \omega(Lu, Lv) = \omega(u, v). }$

Fixing a basis for $\displaystyle{ V }$, $\displaystyle{ \omega }$ can be written as a matrix $\displaystyle{ \Omega }$ and $\displaystyle{ L }$ as a matrix $\displaystyle{ M }$. The condition that $\displaystyle{ L }$ be a symplectic transformation is precisely the condition that M be a symplectic matrix:

$\displaystyle{ M^\text{T} \Omega M = \Omega. }$

Under a change of basis, represented by a matrix A, we have

$\displaystyle{ \Omega \mapsto A^\text{T} \Omega A }$
$\displaystyle{ M \mapsto A^{-1} M A. }$

One can always bring $\displaystyle{ \Omega }$ 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 $\displaystyle{ \Omega }$. As explained in the previous section, $\displaystyle{ \Omega }$ 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 $\displaystyle{ \Omega }$ given above is the block diagonal form

$\displaystyle{ \Omega = \begin{bmatrix} \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{bmatrix}. }$

This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation $\displaystyle{ J }$ is used instead of $\displaystyle{ \Omega }$ 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 $\displaystyle{ \Omega }$ but represents a very different structure. A complex structure $\displaystyle{ J }$ is the coordinate representation of a linear transformation that squares to $\displaystyle{ -I_n }$, whereas $\displaystyle{ \Omega }$ is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which $\displaystyle{ J }$ is not skew-symmetric or $\displaystyle{ \Omega }$ does not square to $\displaystyle{ -I_n }$.

Given a hermitian structure on a vector space, $\displaystyle{ J }$ and $\displaystyle{ \Omega }$ are related via

$\displaystyle{ \Omega_{ab} = -g_{ac}{J^c}_b }$

where $\displaystyle{ g_{ac} }$ is the metric. That $\displaystyle{ J }$ and $\displaystyle{ \Omega }$ 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.

## Diagonalization and decomposition

• For any positive definite symmetric real symplectic matrix S there exists U in $\displaystyle{ \mathrm{U}(2n,\mathbb{R}) = \mathrm{O}(2n) }$ such that
$\displaystyle{ S = U^\text{T} D U \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}), }$
where the diagonal elements of D are the eigenvalues of S.[4]
$\displaystyle{ S = UR \quad }$ for $\displaystyle{ \quad U \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{U}(2n,\mathbb{R}) }$ and $\displaystyle{ R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}). }$
• Any real symplectic matrix can be decomposed as a product of three matrices:

$\displaystyle{ S = O\begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}O', }$

(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.

## Complex matrices

If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors [6] adjust the definition above to

$\displaystyle{ M^* \Omega M = \Omega\,. }$

(3)

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 [7] retain the definition (1) for complex matrices and call matrices satisfying (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.[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]