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 matrices.
...
, 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 ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument s ...
, 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
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner prod ...
.
[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
*
Symplectic spinor bundle
*
Symplectic vector space In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument s ...
Notes
References
*da Silva, A.C.,
Lectures on Symplectic Geometry', Springer (2001). .
*Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006) Birkhäuser Verlag, Basel .
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Symplectic geometry