Canonical Forms for Matrices over Polynomial Rings
Date
2017-09-20
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ORCID
Type
Thesis
Degree Level
Masters
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.
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Keywords
rank of polynomial matrices in more than one indeterminate, lower and upper bounds for ranks of polynomial matrices, simpler or canonical forms for polynomial matrices
Citation
Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Mathematics