Remark 2

Tag 075M, permanent. Cite it as [Bourbaki, Tag 075M].

By the above proposition, there exists a basis of E with respect to which the matrix U of u has the form

\begin{pmatrix} A_{n-r+1} & 0 & \ldots & 0 & 0 \\ 0 & A_{n-r+2} & \ldots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & \ldots & A_{n-1} & 0 \\ 0 & 0 & \ldots & 0 & A_n \end{pmatrix}

where each matrix $A_i$ has the form (2) (taking $g(X) = q_i(X)$ ) (cf. VII, pp. 29-30).

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

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