Symplectic vector space

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.