Symplectic basis

In linear algebra, a standard symplectic basis is a basis ${𝐞}_i,{𝐟}_i$ of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form $\omega$, such that $\omega ({𝐞}_i,{𝐞}_j)=0=\omega ({𝐟}_i,{𝐟}_j),\omega ({𝐞}_i,{𝐟}_j)={\delta }_{ij}$. 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

References

• da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). .
• Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel .