Corollary 2

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

Algebra, chapter IX, § 6. Propriétés spéciales aux formes hermitiennes

*Let $\mathbf{A}$ be a field and $R$ a hermitian matrix of order $n$ and of rank $r$ over $\mathbf{A}$ . Then, unless the following condition is satisfied:

(C') $\mathbf{A}$ is commutative of characteristic 2, $\mathbf{J}$ is the identity, $R$ is alternating and non-zero,* there exists an invertible matrix $P$ of order $n$ over $A$ such that

tP . R . \overline{P} = \begin{pmatrix} \alpha_1 & 0 \ldots 0 \ldots 0 \\ 0 & \alpha_2 \ldots 0 \ldots 0 \\ \cdots & \cdots \\ 0 & 0 \ldots \alpha_r \ldots 0 \\ 0 & 0 \ldots 0 \ldots 0 \\ \cdots & \cdots \\ 0 & 0 \ldots 0 \ldots 0 \end{pmatrix}

where $\overline{\alpha}_i = \alpha_i \neq 0$ for $i = 1, \ldots, r$ .

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

Cited by

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Cites

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