In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, a standard symplectic basis is a basis
of a
symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form
, such that
. A symplectic basis of a symplectic vector space always exists; it can be constructed by a procedure similar to the
Gram–Schmidt process.
[Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006), p.7 and pp. 12–13] 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
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
*
Symplectic frame bundle In symplectic geometry, the symplectic frame bundle
of a given symplectic manifold (M, \omega)\, is the canonical principal (n,)-subbundle \pi_\colon\to M\, of the tangent frame bundle \mathrm FM\, consisting of linear frames which are symplectic ...
*
Symplectic spinor bundle In differential geometry, given a metaplectic structure \pi_\colon\to M\, on a 2n-dimensional symplectic manifold (M, \omega),\, the symplectic spinor bundle is the Hilbert space bundle \pi_\colon\to M\, associated to the metaplectic structure via ...
*
Symplectic vector space
Notes
References
*da Silva, A.C.,
Lectures on Symplectic Geometry', Springer (2001). .
*Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006) Birkhäuser Verlag, Basel .
{{linear-algebra-stub
Symplectic geometry