Corollary 6

Tag 076C, permanent. Cite it as [Bourbaki, Tag 076C].

Algebra, chapter VII, § 5. Endomorphisms of vector spaces

— Suppose the field K is of characteristic zero. Then, in the ring $K[[T]]$ of formal power series, we have

- T \frac{d}{dT} \log \det(1 - Tu) = \sum_{m \geq 1} \operatorname{Tr}(u^m) T^m

Let us first of all work in the field $K(T)$ of rational functions, and put $P(T) = \det(I, -T \cdot U)$ , where $U$ is the matrix of $u$ with respect to some basis of $E$ . Then

P(T) = \det(T(T^{-1} \cdot I, -U)) = T^n \chi_U(T^{-1}) ,

so $P'(T)/P(T) = n/T - \chi_u'(T^{-1})/T^2 \chi_u(T^{-1})$ . Moreover, by Cor. 5 we have

\chi_u'(T^{-1})/T \chi_U(T^{-1}) = \operatorname{Tr}((T^{-1} \cdot I_n - U)^{-1})/T = \operatorname{Tr}((I_n - T \cdot U)^{-1}) .

It follows that $-TP'(T)/P(T) = -n + \operatorname{Tr}((I_n - TU)')$ . The corollary is now obtained by expanding each side of this equation as a formal power series.

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

Cited by

Nothing.

Cites

Cor. 5