Corollary 1

Tag 0DL4, permanent. Cite it as [Bourbaki, Tag 0DL4].

Algebra, chapter IX, § 5. Propriétés spéciales aux formes bilinéaires alternées

*Let $A$ be a commutative field, $E$ a finite-dimensional vector space of dimension $n$ over $A$ , and $\Phi$ an alternating bilinear form on E. There then exists a basis $(e_i)_{1 \leq i \leq n}$ of E and a pair integer $2r \leq n$ such that

\Phi\left( \sum_{i=1}^n \xi_i e_i, \sum_{i=1}^n \eta_i e_i \right) = \sum_{j=1}^r (\xi_{2j-1} \eta_{2j} - \xi_{2j} \eta_{2j-1}).

In particular $\Phi$ is of pair rank $2r$ .

A basis satisfying (2) is called a symplectic basis for $\Phi$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.