Editing Symplectic vector space (section)

==Standard symplectic space==
{{Further|Symplectic matrix#Symplectic transformations}}

The standard symplectic space is <math>\mathbb{R}^{2n}</math> with the symplectic form given by a [[nonsingular matrix|nonsingular]], [[skew-symmetric matrix]]. Typically <math>\omega</math> is chosen to be the [[block matrix]]

:<math>\omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}</math>

where ''I''<sub>''n''</sub> is the {{nowrap|''n'' × ''n''}} [[identity matrix]]. In terms of basis vectors {{nowrap|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'', ''y''<sub>1</sub>, ..., ''y<sub>n</sub>'')}}:

:<math>\begin{align}
  \omega(x_i, y_j) = -\omega(y_j, x_i) &= \delta_{ij}, \\
  \omega(x_i, x_j) =  \omega(y_i, y_j) &= 0.
\end{align}</math>

A modified version of the [[Gram–Schmidt process]] shows that any finite-dimensional symplectic vector space has a basis such that <math>\omega</math> takes this form, often called a '''''Darboux basis''''' or [[symplectic basis]].

'''Sketch of process:''' 

Start with an arbitrary basis <math>v_1, ..., v_n</math>, and represent the dual of each basis vector by the [[dual basis]]: <math>\omega(v_i, \cdot) = \sum_j \omega(v_i, v_j) v_j^*</math>. This gives us a <math>n\times n</math> matrix with entries <math>\omega(v_i, v_j)</math>. Solve for its null space. Now for any <math>(\lambda_1, ..., \lambda_n)</math> in the null space, we have <math>\sum_i \omega(v_i, \cdot) = 0</math>, so the null space gives us the degenerate subspace <math>V_0</math>. 

Now arbitrarily pick a complementary <math>W</math> such that <math>V = V_0 \oplus W</math>, and let <math>w_1, ..., w_m</math> be a basis of <math>W</math>. Since <math>\omega(w_1, \cdot) \neq 0</math>, and <math>\omega(w_1, w_1) = 0</math>, WLOG <math>\omega(w_1, w_2 ) \neq 0</math>. Now scale <math>w_2</math> so that <math>\omega(w_1, w_2) =1</math>. Then define <math>w' = w - \omega(w, w_2) w_1 + \omega(w, w_1) w_2</math> for each of <math>w = w_3, w_4, ..., w_m</math>. Iterate.

Notice that this method applies for symplectic vector space over any field, not just the field of real numbers.

'''Case of real or complex field:'''

When the space is over the field of real numbers, then we can modify the modified Gram-Schmidt process as follows: Start the same way. Let <math>w_1, ..., w_m</math> be an orthonormal basis (with respect to the usual inner product on <math>\R^n</math>)  of <math>W</math>. Since <math>\omega(w_1, \cdot) \neq 0</math>, and <math>\omega(w_1, w_1) = 0</math>, WLOG <math>\omega(w_1, w_2 ) \neq 0</math>. Now multiply <math>w_2</math> by a sign, so that <math>\omega(w_1, w_2) \geq 0</math>. Then define <math>w' = w - \omega(w, w_2) w_1 + \omega(w, w_1) w_2</math> for each of <math>w = w_3, w_4, ..., w_m</math>, then scale each <math>w'</math> so that it has norm one. Iterate.

Similarly, for the field of complex numbers, we may choose a unitary basis. This proves the [[Skew-symmetric matrix#Spectral theory|spectral theory of antisymmetric matrices]].

=== Lagrangian form ===
There is another way to interpret this standard symplectic form. Since the model space '''R'''<sup>2''n''</sup> used above carries much canonical structure which might easily lead to misinterpretation, we will use "anonymous" vector spaces instead. Let ''V'' be a real vector space of dimension ''n'' and ''V''<sup>∗</sup> its [[dual space]]. Now consider the [[direct sum of vector spaces|direct sum]] {{nowrap|1=''W'' = ''V'' ⊕ ''V''<sup>∗</sup>}} of these spaces equipped with the following form:

:<math>\omega(x \oplus \eta, y \oplus \xi) = \xi(x) - \eta(y).</math>

Now choose any [[Basis (linear algebra)|basis]] {{nowrap|(''v''<sub>1</sub>, ..., ''v''<sub>''n''</sub>)}} of ''V'' and consider its [[dual space|dual basis]]

:<math>\left(v^*_1, \ldots, v^*_n\right).</math>

We can interpret the basis vectors as lying in ''W'' if we write {{nowrap|1=''x''<sub>''i''</sub> = (''v''<sub>''i''</sub>, 0) and ''y''<sub>''i''</sub> = (0, ''v''<sub>''i''</sub><sup>∗</sup>)}}. Taken together, these form a complete basis of ''W'',

:<math>(x_1, \ldots, x_n, y_1, \ldots, y_n).</math>

The form ''ω'' defined here can be shown to have the same properties as in the beginning of this section. On the other hand, every symplectic structure is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''<sup>∗</sup>}}. The subspace ''V'' is not unique, and a choice of subspace ''V'' is called a '''polarization'''. The subspaces that give such an isomorphism are called '''Lagrangian subspaces''' or simply '''Lagrangians'''.

Explicitly, given a Lagrangian subspace [[#Subspaces|as defined below]], then a choice of basis {{nowrap|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} defines a dual basis for a complement, by {{nowrap|1=''ω''(''x''<sub>''i''</sub>, ''y''<sub>''j''</sub>) = ''δ''<sub>''ij''</sub>}}.

===Analogy with complex structures===
Just as every symplectic structure is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''<sup>∗</sup>}}, every [[linear complex structure|''complex'' structure]] on a vector space is isomorphic to one of the form {{nowrap|''V'' ⊕ ''V''}}. Using these structures, the [[tangent bundle]] of an ''n''-manifold, considered as a 2''n''-manifold, has an [[almost complex structure]], and the [[cotangent bundle|''co''tangent bundle]] of an ''n''-manifold, considered as a 2''n''-manifold, has a symplectic structure: {{nowrap|1=''T''<sub>∗</sub>(''T''<sup>∗</sup>''M'')<sub>''p''</sub> = ''T''<sub>''p''</sub>(''M'') ⊕ (''T''<sub>''p''</sub>(''M''))<sup>∗</sup>}}.

The complex analog to a Lagrangian subspace is a [[real subspace|''real'' subspace]], a subspace whose [[complexification]] is the whole space: {{nowrap|1=''W'' = ''V'' ⊕ ''J'' ''V''}}. As can be seen from the standard symplectic form above, every symplectic form on '''R'''<sup>2''n''</sup> is isomorphic to the imaginary part of the standard complex (Hermitian) inner product on '''C'''<sup>''n''</sup> (with the convention of the first argument being anti-linear).