 Smith normal form

For every integer matrix $A$, there are unimodular matrices $U,V$ such thatThe positive diagonal entries are unique. Their products $d_1\cdots d_k$ are the greatest common divisors of the $k$-rowed minors.

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.