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

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 say that this matrix must be skew-symmetric, nonsingular, and hollow. 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, hollow matrix of odd size has determinant zero. Notice 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

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