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| In mathematics, a '''symplectic matrix''' is a 2''n''×2''n'' [[matrix (mathematics)|matrix]] ''M'' with [[real number|real]] entries that satisfies the condition
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| {{NumBlk|:|<math>M^T \Omega M = \Omega\,.</math>|{{EquationRef|1}}}}
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| where ''M<sup>T</sup>'' denotes the [[transpose]] of ''M'' and Ω is a fixed 2''n''×2''n'' [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]]. This definition can be extended to 2''n''×2''n'' matrices with entries in other [[field (mathematics)|field]]s, e.g. the [[complex number]]s.
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| Typically Ω is chosen to be the [[block matrix]]
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| :<math>\Omega =
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| \begin{bmatrix}
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| 0 & I_n \\
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| -I_n & 0 \\
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| \end{bmatrix}</math>
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| where ''I''<sub>n</sub> is the ''n''×''n'' [[identity matrix]]. The matrix Ω has [[determinant]] +1 and has an inverse given by Ω<sup>−1</sup> = Ω<sup>''T''</sup> = −Ω.
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| Every symplectic matrix has unit determinant, and the 2''n''×2''n'' symplectic matrices with real entries form a [[subgroup]] of the [[special linear group]] SL(2''n'', ''R'') under [[matrix multiplication]], specifically a [[connected space|connected]] [[compact space|noncompact]] [[real Lie group]] of real dimension {{nowrap|''n''(2''n'' + 1)}}, the [[symplectic group]] Sp(2''n'', '''R'''). The symplectic group can be defined as the set of [[linear transformations]] that preserve the symplectic form of a real [[symplectic vector space]].
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| ==Properties==
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| Every symplectic matrix is [[invertible matrix|invertible]] with the inverse matrix given by
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| :<math>M^{-1} = \Omega^{-1} M^T \Omega.</math>
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| Furthermore, the [[matrix multiplication|product]] of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a [[group (mathematics)|group]]. There exists a natural [[manifold]] structure on this group which makes it into a (real or complex) [[Lie group]] called the [[symplectic group]]. The symplectic group has dimension ''n''(2''n'' + 1).
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| 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. One way to see this is through the use of the [[Pfaffian]] and the identity
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| :<math>\mbox{Pf}(M^T \Omega M) = \det(M)\mbox{Pf}(\Omega).</math> | |
| Since <math>M^T \Omega M = \Omega</math> and <math>\mbox{Pf}(\Omega) \neq 0</math> we have that det(''M'') = 1.
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| Suppose Ω is given in the standard form and let ''M'' be a 2''n''×2''n'' [[block matrix]] given by
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| :<math>M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}</math>
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| where ''A, B, C, D'' are ''n''×''n'' matrices. The condition for ''M'' to be symplectic is equivalent to the conditions
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| :<math>A^TD - C^TB = I</math>
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| :<math>A^TC = C^TA</math>
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| :<math>D^TB = B^TD.</math>
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| When ''n'' = 1 these conditions reduce to the single condition det(''M'') = 1. Thus a 2×2 matrix is symplectic [[iff]] it has unit determinant.
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| With Ω in standard form, the inverse of ''M'' is given by
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| :<math>M^{-1} = \Omega^{-1} M^T \Omega=\begin{pmatrix}D^T & -B^T \\-C^T & A^T\end{pmatrix}.</math>
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| ==Symplectic transformations==
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| In the abstract formulation of [[linear algebra]], matrices are replaced with [[linear transformation]]s 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 2''n''-dimensional vector space ''V'' equipped with a [[nondegenerate form|nondegenerate]], [[skew-symmetric]] [[bilinear form]] ω called the [[symplectic form]].
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| A symplectic transformation is then a linear transformation ''L'' : ''V'' → ''V'' which preserves ω, i.e.
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| :<math>\omega(Lu, Lv) = \omega(u, v).</math>
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| Fixing a [[basis (linear algebra)|basis]] for ''V'', ω can be written as a matrix Ω and ''L'' as a matrix ''M''. The condition that ''L'' be a symplectic transformation is precisely the condition that ''M'' be a symplectic matrix:
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| :<math>M^T \Omega M = \Omega.</math>
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| Under a [[change of basis]], represented by a matrix ''A'', we have
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| :<math>\Omega \mapsto A^T \Omega A</math>
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| :<math>M \mapsto A^{-1} M A.</math>
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| One can always bring Ω to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of ''A''.
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| ==The matrix Ω==
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| Symplectic matrices are defined relative to a fixed [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]] Ω. As explained in the previous section, Ω can be thought of as the coordinate representation of a [[nondegenerate form|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]].
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| The most common alternative to the standard Ω given above is the [[block diagonal]] form
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| :<math>\Omega = \begin{bmatrix}
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| \begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\
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| & \ddots & \\
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| 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix}
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| \end{bmatrix}.</math>
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| This choice differs from the previous one by a [[permutation]] of basis vectors.
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| Sometimes the notation ''J'' is used instead of Ω for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a [[linear complex structure|complex structure]], which often has the same coordinate expression as Ω but represents a very different structure. A complex structure ''J'' is the coordinate representation of a linear transformation that squares to −1, whereas Ω is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which ''J'' is not skew-symmetric or Ω does not square to −1.
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| Given a [[hermitian structure]] on a vector space, ''J'' and Ω are related via
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| :<math>\Omega_{ab} = -g_{ac}{J^c}_b</math>
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| where <math>g_{ac}</math> is the [[metric tensor|metric]]. That ''J'' and Ω 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.
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| ==Complex matrices==
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| If instead ''M'' is a ''2n''×''2n'' [[matrix (mathematics)|matrix]] with [[complex number|complex]] entries, the definition is not standard throughout the literature. Many authors <ref>{{cite journal|last = Xu|first= H. G.|title= An SVD-like matrix decomposition and its applications|journal= Linear Algebra and its Applications|date= July 15, 2003|volume= 368|pages=1–24|doi = 10.1016/S0024-3795(03)00370-7}}</ref> adjust the definition above to
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| {{NumBlk|:|<math>M^* \Omega M = \Omega\,.</math>|{{EquationRef|2}}}}
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| where ''M<sup>*</sup>'' 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.
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| Other authors <ref>{{Cite journal|last1=Mackey |last2= Mackey|first1= D. S. |first2= N.|title= On the Determinant of Symplectic Matrices|year= 2003
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| |series=Numerical Analysis Report| volume= 422|publisher=Manchester Centre for Computational Mathematics|location=Manchester, England
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| }}</ref> retain the definition ({{EquationNote|1}}) for complex matrices and call matrices satisfying ({{EquationNote|2}}) ''conjugate symplectic''.
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| ==See also==
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| {{Portal|Mathematics}}
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| * [[symplectic vector space]]
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| * [[symplectic group]]
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| * [[symplectic representation]]
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| * [[orthogonal matrix]]
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| * [[unitary matrix]]
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| * [[Hamiltonian mechanics]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{planetmath reference|id=4140|title=Symplectic matrix}}
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| * {{planetmath reference|id=7455|title=The characteristic polynomial of a symplectic matrix is a reciprocal polynomial}}
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| [[Category:Matrices]]
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| [[Category:Symplectic geometry]]
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