Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose I know that two vectors $\vec{a}$ and $\vec{b}$ are perpendicular in a given basis spanned by basis vectors $\vec{x}$. Now suppose I transform to another basis $\vec{x'}$ using a symplectic transformation matrix S (i.e. $SJS^{T} = J$ for some skew-symmetric matrix J). Will the transformed vectors a and b still be perpendicular after the transformation? If not, is there a way to figure out a relation between the dot products between the two vectors in the two bases?


share|cite|improve this question
I only can add this part of information: $\langle S a,S b\rangle= a^TS^TSb $ if the vectors are considered as one-column matrices. $S$ need not preserve orthogonality. – Berci Mar 10 '13 at 21:05

Perhaps I misunderstand your question, but here is an example that shows the answer is no to what I think you are asking.

Consider $\mathbb{R}^2$ with the standard symplectic structure given by $\omega( (a,b), (a',b')) = ab' - ba'$. This corresponds to saying $\omega = < J \cdot, \cdot>$ where $J$ is the almost complex structure given by $$ \begin{pmatrix} 0 & -1\\1 & 0 \end{pmatrix}$$

An easy symplectic and orthogonal basis for $\mathbb{R}^2$ is given by $e_1 = (1,0)$ and $e_2 = (0,1)$. A symplectic transformation is given by $e_1 \mapsto e_1 + e_2, e_2 \mapsto e_2$. (i.e. in matrix form $ \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}. )$ The image of this basis is clearly no longer orthogonal.

share|cite|improve this answer

The answer is no in general. For a general (nonlinear) symplectic transformation, the vectors will be pushed-forward by the Jacobian of the transformation. We also know that jacobian of a symplectic transformation is a symplectic linear transformation. So in general, if what you propose is true, then it should hold for ALL symplectic transformations, and not just linear ones like you propose.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.