Canonical Forms for Matrices over Polynomial Rings

Abstract

One of the important concepts in matrix algebra is rank of matrices. If the entries of such matrices are from fields or principal ideal domains, then this concept of rank is well-defined. However, when such matrices are defined over the ring of polynomials F[x_1, . . . , x_k ], k ≥ 2 (polynomial matrices in more than one indeterminate), the concept of rank has different but inequivalent definitions. Despite this flaw, some theories, in relation to ranks, can still be applied to polynomial matrices in more than one indeterminate. One of the outcomes of these theories is that lower and upper bounds for ranks of such polynomial matrices in more than one indeterminate can be obtained. Just like matrices over fields or principal ideal domains can be reduced to some simpler or canonical forms, there are algorithms that can be used to reduce matrices over polynomial rings in more than one indeterminate to some simpler forms, though these reduced forms do not always tell the ranks of such polynomial matrices in more than one indeterminate. In this thesis, these algorithms will be presented with examples.