Symplectic matrix

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

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

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

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

Every symplectic matrix is invertible with the inverse matrix given by $${\displaystyle M^{-1}=\Omega ^{-1}M^{\text{T}}\Omega .}$${\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.

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 $${\displaystyle {\mbox{Pf}}(M^{\text{T}}\Omega M)=\det(M){\mbox{Pf}}(\Omega ).}$${\displaystyle {\mbox{Pf}}(M^{\text{T}}\Omega M)=\det(M){\mbox{Pf}}(\Omega ).} Since ${\displaystyle M^{\text{T}}\Omega M=\Omega }${\displaystyle M^{\text{T}}\Omega M=\Omega } and ${\displaystyle {\mbox{Pf}}(\Omega )\neq 0}${\displaystyle {\mbox{Pf}}(\Omega )\neq 0} we have that ${\displaystyle \det(M)=1}${\displaystyle \det(M)=1}.

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

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

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

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

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

The second condition comes from the fact that if ${\displaystyle M}${\displaystyle M} is symplectic, then ${\displaystyle M^{T}}${\displaystyle M^{T}} is also symplectic. When ${\displaystyle n=1}${\displaystyle n=1} these conditions reduce to the single condition ${\displaystyle \det(M)=1}${\displaystyle \det(M)=1}. Thus a ${\displaystyle 2\times 2}${\displaystyle 2\times 2} matrix is symplectic iff it has unit determinant.

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

A symplectic transformation is then a linear transformation ${\displaystyle L:V\to V}${\displaystyle L:V\to V} which preserves ${\displaystyle \omega }${\displaystyle \omega }, i.e. $${\displaystyle \omega (Lu,Lv)=\omega (u,v).}$${\displaystyle \omega (Lu,Lv)=\omega (u,v).} Fixing a basis for ${\displaystyle V}${\displaystyle V}, ${\displaystyle \omega }${\displaystyle \omega } can be written as a matrix ${\displaystyle \Omega }${\displaystyle \Omega } and ${\displaystyle L}${\displaystyle L} as a matrix ${\displaystyle M}${\displaystyle M}. The condition that ${\displaystyle L}${\displaystyle L} be a symplectic transformation is precisely the condition that M be a symplectic matrix: $${\displaystyle M^{\text{T}}\Omega M=\Omega .}$${\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 \Omega \mapsto A^{\text{T}}\Omega A} $${\displaystyle M\mapsto A^{-1}MA.}$${\displaystyle M\mapsto A^{-1}MA.} One can always bring ${\displaystyle \Omega }${\displaystyle \Omega } 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 ${\displaystyle \Omega }${\displaystyle \Omega }. As explained in the previous section, ${\displaystyle \Omega }${\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 ${\displaystyle \Omega }${\displaystyle \Omega } given above is the block diagonal form $${\displaystyle \Omega ={\begin{bmatrix}{\begin{matrix}0&1\\-1&0\end{matrix}}&&0\\&\ddots &\0円&&{\begin{matrix}0&1\\-1&0\end{matrix}}\end{bmatrix}}.}$${\displaystyle \Omega ={\begin{bmatrix}{\begin{matrix}0&1\\-1&0\end{matrix}}&&0\\&\ddots &\0円&&{\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 ${\displaystyle J}${\displaystyle J} is used instead of ${\displaystyle \Omega }${\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 ${\displaystyle \Omega }${\displaystyle \Omega } but represents a very different structure. A complex structure ${\displaystyle J}${\displaystyle J} is the coordinate representation of a linear transformation that squares to ${\displaystyle -I_{n}}${\displaystyle -I_{n}}, whereas ${\displaystyle \Omega }${\displaystyle \Omega } is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which ${\displaystyle J}${\displaystyle J} is not skew-symmetric or ${\displaystyle \Omega }${\displaystyle \Omega } does not square to ${\displaystyle -I_{n}}${\displaystyle -I_{n}}.

Given a hermitian structure on a vector space, ${\displaystyle J}${\displaystyle J} and ${\displaystyle \Omega }${\displaystyle \Omega } are related via $${\displaystyle \Omega _{ab}=-g_{ac}{J^{c}}_{b}}$${\displaystyle \Omega _{ab}=-g_{ac}{J^{c}}_{b}} where ${\displaystyle g_{ac}}${\displaystyle g_{ac}} is the metric. That ${\displaystyle J}${\displaystyle J} and ${\displaystyle \Omega }${\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.

If instead M is a 2n ×ばつ 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 ×ばつ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.^[12]

• Symplectic vector space
• Symplectic group
• Symplectic representation
• Orthogonal matrix
• Unitary matrix
• Hamiltonian mechanics
• Linear complex structure
• Williamson theorem
• Hamiltonian matrix

• Matrices (mathematics)
• Symplectic geometry

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