Symplectic basis

In linear algebra, a standard symplectic basis is a basis formula_1 of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form formula_2, such that formula_3. 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.