Symplectic basis theorem - Imported Knowlpedia Content

Statement

Let $(V,\omega)$ be a finite-dimensional real symplectic vector space. The symplectic basis theorem states that $\dim V=2n$ for some $n$ and that $V$ has a symplectic basis

(e_1,\ldots,e_n,f_1,\ldots,f_n)

with

\omega(e_i,e_j)=\omega(f_i,f_j)=0, \qquad \omega(e_i,f_j)=\delta_{ij}.

Consequently every nondegenerate alternating form of dimension $2n$ has the same matrix after a change of basis. In particular, no odd-dimensional real vector space admits a symplectic form.

Proof idea

Choose a nonzero vector $e_1$ . Nondegeneracy supplies $f_1$ with $\omega(e_1,f_1)=1$ . The plane $H_1=\operatorname{span}(e_1,f_1)$ is symplectic, and

V=H_1\oplus H_1^\omega.

The restriction of $\omega$ to $H_1^\omega$ is again nondegenerate. Induction repeats the construction until the complement is zero. This is the alternating-form analogue of Gram–Schmidt orthogonalization Cannas da Silva, §1.1.

Consequences

The decomposition into symplectic planes proves that the dimension is even. It also shows that two symplectic vector spaces over the same field are linearly symplectomorphic exactly when they have the same dimension. In particular, finite-dimensional symplectic linear algebra has no signature invariant analogous to that of a real symmetric bilinear form.

Hypotheses and terminology

The same statement holds over any field for a nondegenerate alternating form. Characteristic different from $2$ is needed only when one replaces “alternating” by “skew-symmetric”; in characteristic $2$ , those conditions are not equivalent. The finite-dimensional hypothesis is essential: an infinite-dimensional topological symplectic space need not possess a basis adapted to its form. “Linear Darboux theorem” refers to this algebraic result, not to the local-coordinate theorem for symplectic manifolds.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, the symplectic basis construction.
Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. Publisher record. Relevant: Chapter 1, canonical form of a nondegenerate alternating form.