Symplectic Vector Space

In mathematics, a symplectic vector space is a vector space V (over a field, for example the real numbers R) equipped with a bilinear form ω : V × V → R that is

• Skew-symmetric: ω(u, v) = −ω(v, u) for all u, v ∈ V.
• Totally isotropic ω(v, v) = 0 for all v ∈ V.
• Nondegenerate: if ω(u, v) = 0 for all v ∈ V then u = 0.

The bilinear form ω is said to be a symplectic form in this case.

In practice, the above three properties (skew-symmetric, isotropic and nondegenerate) need not all be checked to see that some bilinear form is symplectic:

• The skew-symmetric property is redundant (as a condition), as it follows from the isotropic property (applied to v, to u and to v+u and then combined). Hence, the skew-symmetric property needs not be checked if the isotropic property is known to hold.
• If the underlying field has characteristic ≠ 2, the isotropic property is actually equivalent to the skew-symmetric property. Thus, the isotropic property needs not be checked if the skew-symmetric property is known to hold and the field has characteristic ≠2. On the other hand, if the characteristic is 2, the skew-symmetric property is implied by, but does not imply, the isotropic property. 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 three conditions above say that this matrix must be skew-symmetric and nonsingular. This is not the same thing as 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 matrix of odd size has determinant zero.

A symplectic form behaves quite differently from a symmetric form, such as the dot product on Euclidean vector spaces. With a Euclidean inner product g, we have g(v,v) > 0 for all nonzero vectors v.