Corollary 3

Tag 0DL7, permanent. Cite it as [Bourbaki, Tag 0DL7].

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

Let $A$ be a commutative field, $R$ an alternating square matrix over $A$ . The rank of $R$ is an even number $2r$ , and there exists an invertible matrix $P$ over $A$ such that

{}^tP.R.P = \begin{pmatrix} 0 & I_r & 0 \\ -I_r & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.

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

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