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:M^T \Omega M = \Omega.
where ''MT'' denotes the Transpose of ''M'' and Ω is the ''2n''×''2n'' Skew-symmetric Matrix
:\Omega =
\begin{bmatrix}
0 & I_n \
-I_n & 0 \
\end{bmatrix}
Here ''I''n is the ''n''×''n'' Identity Matrix . Note that Ω has Determinant +1 and squares to minus the identity: Ω2 = −''I''2n.

''N.B.'' Some authors prefer to use a different Ω for the definition of symplectic matrices. The only essential property is that Ω be a Nonsingular , Skew-symmetric Matrix . The most common alternative is the Block Diagonal form
:\Omega = \begin{bmatrix}
\begin{matrix}0 & 1\ -1 & 0\end{matrix} & & 0 \
& \ddots & \
0 & & \begin{matrix}0 & 1 \ -1 & 0\end{matrix}
\end{bmatrix}
Note that this differs from the previous choice by a Permutation of basis vectors. In fact, any choice of Ω can be brought to either of the above forms by a different choice of basis. See the abstract formulation below in the section on symplectic transformations.

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 a Linear Complex Structure , as described below.


PROPERTIES


Every symplectic matrix has an Inverse which is given by
:M^{-1} = \Omega^{-1} M^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 . The symplectic group has dimension ''n''(2''n'' + 1).

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
:\mbox{Pf}(M^T \Omega M) = \det(M)\mbox{Pf}(\Omega).
Since M^T \Omega M = \Omega and \mbox{Pf}(\Omega)
eq 0 we have that det(''M'') = 1.

Let ''M'' be a 2''n''×2''n'' Block Matrix given by
:M = \begin{pmatrix}A & B \ C & D\end{pmatrix}
where ''A, B, C, D'' are ''n''×''n'' matrices. Then the condition for ''M'' to be symplectic is equivalent to the conditions
:A^TD - C^TB = 1
:A^TC = C^TA
:D^TB = B^TD.

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.


SYMPLECTIC TRANSFORMATIONS


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 , Skew-symmetric Bilinear Form ω.

A symplectic transformation is then a linear transformation ''L'' : ''V'' → ''V'' which preserves ω, i.e.
:\omega(Lu, Lv) = \omega(u, v).
Fixing a 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:
:M^T \Omega M = \Omega.

Under a Change Of Basis , represented by a matrix ''A'', we have
:\Omega \mapsto A^T \Omega A
:M \mapsto A^{-1} M A.
One can always bring Ω to either of the standard forms given in the introduction by a suitable choice of ''A''.


NOTATION: J VS. Ω

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 a Linear Complex Structure , which often has the same coordinate expression but represents a very different structure. These are two different things and should be distinguished. In particular, one could easily choose a basis for which Ω2 ≠ −1, whereas this is an essential quality of a complex structure. Moreover, ''J'' should be understood as a linear transformation whereas Ω is a bilinear form.

Given a Hermitian Structure on a vector space, ''J'' and Ω are related via
:\Omega_{ab} = g_{ac}{J^c}_b
where g_{ac} is the Metric . That ''J'' and Ω can sometimes have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric ''g'' is often the identity matrix.


SEE ALSO



REFERENCES