Symplectic vector space

In mathematics, a symplectic vector space is a vector space ${\displaystyle V}$ over a field ${\displaystyle F}$ (for example the real numbers ${\displaystyle \mathbb {R} }$) equipped with a symplectic bilinear form.

A symplectic bilinear form is a mapping ${\displaystyle \omega :V\times V\to F}$ that is

Bilinear

Linear in each argument separately;

Alternating

${\displaystyle \omega (v,v)=0}$ holds for all ${\displaystyle v\in V}$; and

Non-degenerate

${\displaystyle \omega (v,u)=0}$ for all ${\displaystyle v\in V}$ implies that ${\displaystyle u=0}$.

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, ${\displaystyle \omega }$ 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 ${\displaystyle V}$ 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.