Smith Normal Form - Definition

Definition

Let A be a nonzero m×n matrix over a principal ideal domain R. There exist invertible and -matrices S, T so that the product S A T is

\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}.

and the diagonal elements satisfy . This is the Smith normal form of the matrix A. The elements are unique up to multiplication by a unit and are called the elementary divisors, invariants, or invariant factors. They can be computed (up to multiplication by a unit) as

where (called i-th determinant divisor) equals the greatest common divisor of all minors of the matrix A.

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