Symplectic vector space

A symplectic vector space is a pair where is a vector space and is a non-degenerate skew-symmetric bilinear form. In more detail, this means that

Skew-symmetric

Non-degenerate For every ,

Morphisms

A morphism in the category of symplectic vector spaces is a linear symplectomorphism.

Subspaces

For a subspace , we call the subspace

• isotropic if
• coisotropic if
• symplectic if
• Lagrangian if Where denotes the symplectic complement.

Symplectic basis

For a symplectic vector space , there exists a basis such that

This basis is called a symplectic basis for . The bilinear form in this basis is

where and are written in coordinates with respect to the symplectic basis.

Proof

Using the biliear form (that is non-degenerated) you can use a similar process to Gram-Schmidt to find “symplecto-orthogonal” basis.