Editing Symplectic manifold (section)

=== Symplectic vector spaces ===
{{main|Symplectic vector space}}

Let <math>\{v_1, \ldots, v_{2n}\}</math> be a basis for <math>\R^{2n}.</math> We define our symplectic form <math>\omega</math> on this basis as follows:

:<math>\omega(v_i, v_j) = \begin{cases} 1 & j-i =n \text{ with } 1 \leqslant i \leqslant n \\ -1 & i-j =n \text{ with } 1 \leqslant j \leqslant n \\ 0 & \text{otherwise} \end{cases}</math>

In this case the symplectic form reduces to a simple [[quadratic form]]. If <math>I_n</math> denotes the <math>n\times n</math> [[identity matrix]] then the matrix, <math>\Omega</math>, of this quadratic form is given by the <math>2n\times 2n</math> [[block matrix]]:

:<math>\Omega = \begin{pmatrix} 0 & I_n  \\ -I_n & 0 \end{pmatrix}. </math>