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 matrix (mathemat ...
, a standard symplectic basis is a basis
of a
symplectic vector space
In mathematics, a symplectic vector space is a vector space V over a Field (mathematics), field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a map (mathematics), mapping \omega : ...
, 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
*
Symplectic frame bundle
*
Symplectic spinor bundle
*
Symplectic vector space
In mathematics, a symplectic vector space is a vector space V over a Field (mathematics), field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a map (mathematics), mapping \omega : ...
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