Editing Symplectic vector space (section)

{{Short description|Mathematical concept}}
In [[mathematics]], a '''symplectic vector space''' is a [[vector space]] <math>V</math> over a [[Field (mathematics)|field]] <math>F</math> (for example the real numbers <math>\mathbb{R}</math>) equipped with a symplectic [[bilinear form]].

A '''symplectic bilinear form''' is a [[map (mathematics)|mapping]] <math>\omega : V \times V \to F</math> that is
; [[bilinear form|Bilinear]]: [[linear map|Linear]] in each argument separately;
; [[alternating form|Alternating]]: <math>\omega(v, v) = 0</math> holds for all <math>v \in V</math>; and
; [[Nondegenerate form|Non-degenerate]]: <math>\omega(v, u) = 0</math> for all <math>v \in V</math> implies that <math>u = 0</math>.

If the underlying [[field (mathematics)|field]] has [[characteristic (algebra)|characteristic]] not 2, alternation is equivalent to [[skew symmetry|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 bilinear form|symmetric form]], but not vice versa. 

Working in a fixed [[basis (linear algebra)|basis]], <math>\omega</math> can be represented by a [[matrix (mathematics)|matrix]]. The conditions above are equivalent to this matrix being [[skew-symmetric matrix|skew-symmetric]], [[nonsingular matrix|nonsingular]], and [[hollow matrix#Diagonal entries all zero|hollow]] (all diagonal entries are zero). This should not be confused with a [[symplectic matrix]], which represents a symplectic transformation of the space. If <math>V</math> is [[finite-dimensional]], then its dimension must necessarily be [[even number|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.