Täisarvuliste maatriksite Smithi normaalkuju

Abstract

This bachelor’s thesis gives an overview of the Smith normal form for integral matrices, i.e. matrices whose entries are integers. This normal form is a diagonalization that exists for any integral matrix and, moreover, is uniquely determined. It was first used in the 1861 paper by H.J.S. Smith which considered solving linear diophantine equations and congruences. The Smith normal form has seen an extensive number of applications since then, including diophantine analysis, integer programming, linear systems theory and module theory over principal ideal domains. The thesis itself is a review of fundamentals and contains no original research, but it does strive to be as elementary and self-contained as possible. There are altogether three chapters. The first one introduces a number of definitions and results from elementary matrix algebra and number theory that will be needed later on. The second chapter introduces the notion of equivalent matrices. It also contains the main result of the thesis, the proof that every integral matrix has a Smith normal form, i.e. is equivalent to a specific kind of diagonal matrix. There is also a subchapter on certain invariants of equivalent matrices, namely determinantal divisors and invariant factors, which are used to prove that the Smith normal form is unique. Finally, the process of finding the Smith normal form of an integral matrix is illustrated by a simple numerical example. The last chapter contains an overview of three applications: solving linear diophantine equations, a method for analysing a certain class of combinatorial problems and the fundamental theorem of finitely generated Abelian groups.