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 VT1 and VT2 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 VT. 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.KeywordsNormal FormLeft IdealInvariant FactorElementary MatriceElementary DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.