Corollary 1

Tag 0743, permanent. Cite it as [Bourbaki, Tag 0743].

Algebra, chapter VII, § 4. Finitely generated modules over a principal ideal domain

— Let X be a matrix of rank r, with n rows and m columns, over a principal ideal domain A; then there exists a matrix $X_0$ equivalent to X (II, p. 354) of the form

\begin{pmatrix} \alpha_1 & 0 & \ldots & 0 & 0 & \ldots & 0 \\ 0 & \alpha_2 & \ldots & 0 & 0 & \ldots & 0 \\ 0 & 0 & \ldots & \alpha_r & 0 & \ldots & 0 \\ 0 & 0 & \ldots & 0 & 0 & \ldots & 0 \\ 0 & 0 & \ldots & 0 & 0 & \ldots & 0 \end{pmatrix}

where the $\alpha_i$ are nonzero elements of A, each of which divides the next. Under these conditions the $\alpha_i$ are uniquely determined up to multiplication by invertible elements.

Given that two matrices X and $X'$ are equivalent if there exist invertible square matrices P and Q, of orders n and m, over A, such that $X' = PXQ$ , Cor. 1 is just Prop. 5 expressed in terms of matrices.

In the notation of Prop. 5 and Cor. 1, the (nonzero) ideals $Au$ , are called the invariant factors of the linear map f, or of the matrix X. It then follows immediately from Cor. I that:

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

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Corollary 1

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