Remark 5

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

Algebra, chapter IX, § 7. Formes hermitiennes et corps ordonnés

One assumes that $A = K$ or $A = K(i)$ ; let $R, S$ be two hermitian matrices over $A$ , of signatures $(s, t)$ and $(s', t')$ respectively (exerc. 4); show that the matrix $R \otimes S$ is a hermitian matrix of signature $(ss' + tt', st' + s't)$ .

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

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