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{wiki=Smith_normal_form}

The Smith normal form is a canonical form for matrices over integers (or more generally, over any principal ideal domain) that reveals important structural information about the matrix. It is primarily used in the study of finitely generated modules over rings, especially in linear algebra and number theory.


Smith normal form

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For every matrix $A$ over a <principal ideal domain>, there are invertible matrices $U,V$ such that
$$
UAV=\operatorname{diag}(d_1,\ldots,d_r,0,\ldots,0),
\qquad d_1\mid d_2\mid\cdots\mid d_r.
$$
The nonzero diagonal entries are unique up to multiplication by units. Over the <integer>[integers], $U$ and $V$ are unimodular and the $d_i$ may be chosen positive; their products $d_1\cdots d_k$ are the greatest common divisors of the $k$-rowed minors.


Smith normal form

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Smith normal form

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