In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle \mathbb {R} } ) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping ω : V × V → F {\displaystyle \omega :V\times V\to F} that is
Bilinear Linear in each argument separately; Alternating
ω ( v , v ) = 0 {\displaystyle \omega (v,v)=0} holds for all v ∈ V {\displaystyle v\in V} ; and Non-degenerate
ω ( v , u ) = 0 {\displaystyle \omega (v,u)=0} for all v ∈ V {\displaystyle v\in V} implies that u = 0 {\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, ω {\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 V {\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 redundant unless the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form such as the scalar product on Euclidean vector spaces.
Standard symplectic space
The standard symplectic space is R 2 n {\displaystyle \mathbb {R} ^{2n}} with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically ω {\displaystyle \omega } is chosen to be the block matrix
ω = [ 0 I n − I n 0 ] {\displaystyle \omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}
where I n {\displaystyle I_{n}} is the n × n {\displaystyle n\times n} identity matrix. In terms of basis vectors ( x 1 , ⋯ , x n , y 1 , ⋯ , y n ) {\displaystyle (x_{1},\cdots ,x_{n},y_{1},\cdots ,y_{n})} :
ω ( x i , y j ) = − ω ( y j , x i ) = δ i j , ω ( x i , x j ) = ω ( y i , y j ) = 0. {\displaystyle {\begin{aligned}\omega (x_{i},y_{j})=-\omega (y_{j},x_{i})&=\delta _{ij},\\\omega (x_{i},x_{j})=\omega (y_{i},y_{j})&=0.\end{aligned}}}
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