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Concept
Symplectic vector space
Summary
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping ω : V × V → F that is
Bilinear Linear in each argument separately;
Alternating ω(v, v) = 0 holds for all v ∈ V; and
Non-degenerate ω(u, v) = 0 for all v ∈ V implies that u = 0.
If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa.
Working in a fixed basis, ω can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If V is finite-dimensional, then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces.
The standard symplectic space is R2n with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically ω is chosen to be the block matrix
where In is the n × n identity matrix. In terms of basis vectors (x1, ..., xn, y1, ..., yn):
A modified version of the Gram–Schmidt process shows that any finite-dimensional symplectic vector space has a basis such that ω takes this form, often called a Darboux basis or symplectic basis.
Sketch of process:
Start with an arbitrary basis , and represent the dual of each basis vector by the dual basis: . This gives us a matrix with entries . Solve for its null space. Now for any in the null space, we have , so the null space gives us the degenerate subspace .
Official source
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