# Symplectic basis

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Symplectic_basis
> Markdown URL: https://mediated.wiki/source/Symplectic_basis.md
> Source: https://en.wikipedia.org/wiki/Symplectic_basis
> Source revision: 1187632693
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [linear algebra](/source/Linear_algebra), a standard **symplectic basis** is a basis {\mathbf e}_i, {\mathbf f}_i of a [symplectic vector space](/source/Symplectic_vector_space), which is a vector space with a nondegenerate alternating bilinear form \omega, such that \omega({\mathbf e}_i, {\mathbf e}_j) = 0 = \omega({\mathbf f}_i, {\mathbf f}_j), \omega({\mathbf e}_i, {\mathbf f}_j) = \delta_{ij}. A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the [Gram–Schmidt process](/source/Gram%E2%80%93Schmidt_process).[1] The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.

## See also

- [Darboux theorem](/source/Darboux_theorem)
- [Symplectic frame bundle](/source/Symplectic_frame_bundle)
- [Symplectic spinor bundle](/source/Symplectic_spinor_bundle)
- [Symplectic vector space](/source/Symplectic_vector_space)

## Notes

1. Maurice de Gosson: *Symplectic Geometry and Quantum Mechanics* (2006), p.7 and pp. 12–13

## References

- da Silva, A.C., *[Lectures on Symplectic Geometry](https://link.springer.com/book/10.1007/978-3-540-45330-7/)*, Springer (2001). ISBN 3-540-42195-5.
- Maurice de Gosson: *Symplectic Geometry and Quantum Mechanics* (2006) Birkhäuser Verlag, Basel ISBN 978-3-7643-7574-4.

---
Adapted from the Wikipedia article [Symplectic basis](https://en.wikipedia.org/wiki/Symplectic_basis) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Symplectic_basis?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.