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 if and only if

$(1): \quad \mathbf A$ 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.

Also see

• Results about Smith normal form can be found here.

Source of Name

This entry was named for Henry John Stephen Smith.

Sources

• 1861: Henry J. Stephen Smith: On systems of linear indeterminate equations and congruences (Phil. Trans. R. Soc. Vol. 151, no. 1: pp. 293 – 326)  www.jstor.org/stable/108738