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. The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite. 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. The existence of the basis implies in particular that the dimension of a symplectic vector space is even if it is finite.