# Definition of Symplectic Matrix

In Wikipedia and also MathPlanet an equal definition of a symplectic matrix is offered :

$$\left( \begin{array}{ccc} A & B \\ C & D \end{array} \right)$$

is symplectic if and also just if :

$$A^TD-C^TB=I, A^TC=C^TA, D^TB=B^TD$$

yet it appears incorrect, given that, as an example :

$$\left( \begin{array}{ccc} 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right)$$

is symplectic yet does not please the problems. Or have I combined every little thing up?

MODIFY : this is insane talk. That matrix isn't symplectic! (except the kind specified in Wikipedia or MathPlanet.

This is an additional inquiry which highlights the troubles with not thinking of points in a coordinate - free fashion. Symplectic makeovers are specified about a symplectic form, and also symplectic matrices subsequently are specified about some "canonical" symplectic kind relative to the typical basis. The trouble is that there go to the very least 2 practical selections for such a "canonical" kind (both of which are defined at the Wikipedia write-up), and also the resulting symplectic matrices you obtain from each kind are various. So you are possibly simply making use of a various one.