Informally, a symplectic map is a map which preserves the sum of areas projected onto the set of 
planes. It is the generalization of an area-preserving
map.
Formally, a symplectic map is a real-linear map 
that preserves a symplectic
form 
,
i.e., for which
 |
for all 
,
. Every symplectic map 
on a complex Hilbert space 
may be written as 
, where 
is unitary, 
is positive, and 
is an anti-linear involution (i.e., complex conjugation).
