Matrices over PIDs

Abstract

Recall that if T _i : V→V (i = 1,2) are linear transformations on a finite-dimensional vector space V over a field F, then T₁ and T₂ are similar if and only if the F[X]-modules V_T1 and V_T2 are isomorphic (Theorem 4.4.2). Since the structure theorem for finitely generated torsion F[X]-modules gives a criterion for isomorphism in terms of the invariant factors (or elementary divisors), one has a powerful tool for studying linear transformations, up to similarity. Unfortunately, in general it is difficult to obtain the invariant factors or elementary divisors of a given linear transformation. We will approach the problem of computation of invariant factors in this chapter by studying a specific presentation of the F[X]-module V_T. This presentation will be used to transform the search for invariant factors into performing elementary row and column operations on a matrix with polynomial entries. We begin with the following definition.