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
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.
Read more about this topic: Smith Normal Form
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