Symplectic basis

In linear algebra, a standard symplectic basis is a basis ${\displaystyle {\mathbf {e} }_{i},{\mathbf {f} }_{i}}$ of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form ${\displaystyle \omega }$, such that ${\displaystyle \omega ({\mathbf {e} }_{i},{\mathbf {e} }_{j})=0=\omega ({\mathbf {f} }_{i},{\mathbf {f} }_{j}),\omega ({\mathbf {e} }_{i},{\mathbf {f} }_{j})=\delta _{ij}}$. A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the Gram–Schmidt process. 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
• Symplectic frame bundle
• Symplectic spinor bundle
• Symplectic vector space

Notes

References

• da Silva, A.C., Lectures on Symplectic Geometry, 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.