• Source: Symplectic basis

In linear algebra, a standard symplectic basis is a basis






e



i


,



f



i




{\displaystyle {\mathbf {e} }_{i},{\mathbf {f} }_{i}}

of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form



ω


{\displaystyle \omega }

, such that



ω
(



e



i


,



e



j


)
=
0
=
ω
(



f



i


,



f



j


)
,
ω
(



e



i


,



f



j


)
=

δ

i
j




{\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.

Kata Kunci Pencarian:

• Pluto
• Symplectic basis
• Symplectic vector space
• Symplectic matrix
• Symplectic manifold
• Symplectic group
• Darboux basis
• Conjugate variables
• Poisson manifold
• Classical group
• Darboux's theorem