Symplectic basis

In linear algebra, a standard symplectic basis is a basis [math]\displaystyle{ {\mathbf e}_i, {\mathbf f}_i }[/math] of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form [math]\displaystyle{ \omega }[/math], such that [math]\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} }[/math]. 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

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.