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.