Definition:Smith Normal Form

Definition
Let $\mathbf A$ be a non-zero $m \times n$ matrix over a principal ideal domain $R$.

Then $\mathbf A$ is of Smith normal form


 * $(1): \quad \mathbf A$ is a diagonal matrix:


 * $\begin {pmatrix}

\alpha_1 & 0 & 0 & & \cdots & & 0 \\ 0 & \alpha_2 & 0 & & \cdots & & 0 \\ 0 & 0 & \ddots & & & & 0 \\ \vdots & & & \alpha_r & & & \vdots \\ & & & & 0 & & \\ & & & & & \ddots & \\ 0 & & & \cdots & & & 0 \end{pmatrix}$


 * $(2): \quad$ The diagonal elements $\alpha_i$ of $\mathbf A$ satisfy:


 * $\forall i \in \set {1, 2, \ldots, r}: \alpha_i \divides \alpha_{i + 1}$
 * where $\divides$ denotes divisibility.