Editing Symplectic matrix (section)

==The matrix Ω==
Symplectic matrices are defined relative to a fixed [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]] <math>\Omega</math>. As explained in the previous section, <math>\Omega</math> 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]].

The most common alternative to the standard <math>\Omega</math> given above is the [[block diagonal]] form
<math display="block">\Omega = \begin{bmatrix}
\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\
 & \ddots & \\
0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix}
\end{bmatrix}.</math>
This choice differs from the previous one by a [[permutation]] of [[basis vectors]].

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

Given a [[hermitian structure]] on a vector space, <math>J</math> and <math>\Omega</math> are related via
<math display="block">\Omega_{ab} = -g_{ac}{J^c}_b</math>
where <math>g_{ac}</math> is the [[metric tensor|metric]]. That <math>J</math> and <math>\Omega</math> 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.