Bourbaki

Remark 3

Tag 09T8, permanent. Cite it as [Bourbaki, Tag 09T8].

Integration, chapter IX, Appendix 0. Complements on Hilbert spaces

— Assume E finite-dimensional and H nondegenerate. Let (e1,…,en)(e_1, \ldots, e_n) be a basis of E. Set q=((ei∣ej)Q)1≤i,j≤nq = ((e_i|e_j)_Q)_{1 \leq i,j \leq n} and h=((ei∣ej)H)1≤i,j≤nh = ((e_i|e_j)_H)_{1 \leq i,j \leq n}. With notations as in Prop. 1, the matrix of uu for the basis (e1,…,en)(e_1, \ldots, e_n) of E is equal to h−1qh^{-1}q, whence

(2)

Tr⁡(Q/H)=Tr⁡(h−1q)=Tr⁡(qh−1). \operatorname{Tr}(Q/H) = \operatorname{Tr}(h^{-1}q) = \operatorname{Tr}(qh^{-1}).

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

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Transcribed from the printed volume. Not translated.