Definition of Symplectic Matrix

Question

In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given:

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

is symplectic if and only if:

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

but it seems wrong, since, for 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 but doesn't satisfy the conditions. Or have I mixed everything up?

EDIT: this is crazy talk. That matrix isn't symplectic! (not for the form defined in Wikipedia or MathPlanet.

Answer 1

This is another question which highlights the problems with not thinking about things in a coordinate-free manner. Symplectic transformations are defined relative to a symplectic form, and symplectic matrices in turn are defined relative to some "canonical" symplectic form with respect to the standard basis. The problem is that there are at least two reasonable choices for such a "canonical" form (both of which are described at the Wikipedia article), and the resulting symplectic matrices you get from each form are different. So you are probably just using a different one.

Answer 2

A $2n \times 2n$ matrix $X$ is called symplectic iff it satisfies $X^t J X = J$, where

$$J = \begin{bmatrix} 0 & I \\-I & 0\end{bmatrix}$$ with $I$ being an identity matrix of rank $n$.

It is easy to check by partitioning $X$ into block form $X=\begin{bmatrix}A & B \\C & D\end{bmatrix}$ that this definition is identical to the one mentioned in both Wikipedia and MathPlanet. So the definition is OK!

But the matrix in your case does not satisfy the equation $X^t JX=J$. I strongly recommend that you check it directly.

Cheers！

Richard