Let $V$ be a symplectic vector space, i.e. a vector space with a non-degenerate alternating bilinear form, a so-called symplectic form. There is a theorem:

All symplectic forms on $V$ are ismorphic.

I have two questions about this:

1) Can somebody explain what it means precisely that two symplectic forms are isomorphic?

2) Could somebode give references on this statement with or without proof?

Thank you very much.


It can be proven that any finite-dimensional symplectic vector space of dimension $2n$ has a basis in which the symplectic form $\omega$ has the form:

$$\begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$$

This proves that two symplectic forms are isomorphic. A reference is Wikipedia.

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  • $\begingroup$ Merci! And what does it mean if two such forms are isomorphic? $\endgroup$ – user586527 Aug 27 '18 at 16:20
  • $\begingroup$ It means that for any $\omega_1,\omega_2$ two symplectic forms, you can find an invertible linear transformation $\varphi$ such that for any $x,y \in V$ $\omega_1(x,y) = \omega_2(\varphi(x),\varphi(y))$. $\endgroup$ – mathcounterexamples.net Aug 27 '18 at 16:24
  • $\begingroup$ So, "Two symplectic vector-spaces of equal dimension are isomorphic" is right as well? $\endgroup$ – user586527 Aug 27 '18 at 16:46
  • $\begingroup$ Ate isomorphic as symplectic spaces. Yes. Any two real Vector spaces of same dimensions are isomorphic in general. $\endgroup$ – mathcounterexamples.net Aug 27 '18 at 16:49
  • 1
    $\begingroup$ What does ate mean? $\endgroup$ – user586527 Aug 27 '18 at 16:50

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