Symplectic vector space

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.