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.

0
2019-12-03 23:51:50
Source Share
Answers: 1

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.

0
2019-12-05 03:00:27
Source