Symplectic basis

In linear algebra, a standard symplectic basis is a basis {\mathbf e}_i, {\mathbf f}_i of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega, such that \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.[1] The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.

See also


  1. Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006), p.7 and pp. 12–13


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