The Smith Normal Form of a Matrix

In this note we will discuss the structure theorem for finitely generated modules over a principal ideal domain from the point of view of matrices. We will then give a matrixtheoretic proof of the structure theorem from the point of view of the Smith normal form of a matrix over a principal ideal domain. One benefit from this method is that there are algorithms for finding the Smith normal form of a matrix, and these are programmed into common computer algebra packages such as Maple and MuPAD. These packages will make it easy to decompose a finitely generated module over a polynomial ring F [x] into a direct sum of cyclic submodules. To start, we will need to discuss describing a module by generators and relations. To motivate the definition, let F be a field, and take A ∈ Mn(F ). We can make F , viewed as the set of column matrices over F , into an F [x]-module by defining f(x)v = f(A)v. This module structure is dependent on A; we denote this module by (F ). Write A = (aij). If {e1, . . . , en} is the standard basis of F , then xej = Aej = ∑n i=1 aijei for each j. Consequently,