Polinomijalne matrice

master's thesis

• Author: Matea Vujičić
• Mentor: Maja Starčević (mentor)
• Committee member: Maja Starčević (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom smo radu opisali neke osnovne pojmove, teoreme i rezultate vezane uz polinomijalne matrice. Najprije smo definirali poznate pojmove poput polinoma, matrice i osnovnih operacija nad njima, a nakon toga smo uveli definiciju polinomijalne matrice i osnovnih operacija nad polinomijalnim matricama. Opisali smo operacije zbrajanja, množenja i množenja skalarom, a opisani su mnogi važni teoremi vezani uz dijeljenje, između ostalog i mali Bézoutov teorem koji smo poopćili. Također smo vidjeli mnoge primjere koji potkrepljuju dokazane teoreme te algoritme koji proizlaze iz tih teorema. Objasnili smo linearnu nezavisnost polinomijalnih vektora te definirali baze za prostor takvih vektora i rang polinomijalnih matrica. Vidjeli smo da su sve te definicije analogne onima u klasičnom vektorskom prostoru gdje nemamo polinomijalne vektore. Naveli smo primjere linearno nezavisnih, kao i linearno zavisnih polinomijalnih vektora. Uveli smo važan pojam Smithove kanonske forme te invarijantnih polinoma koje smo koristili i u narednim cjelinama. Definirali smo ekvivalenciju polinomijalnih matrica te pokazali vezu između sličnosti klasičnih matrica i pripadnih polinomijalnih matrica prvog stupnja. Od novih pojmova smo također vidjeli što su to elementarni djelitelji te nultočke polinomijalne matrice, a sve to zapravo proizlazi iz definicije Smithove kanonske forme i definicije invarijantnih polinoma. Na samom kraju objasnili smo Frobeniusovu kanonsku formu polinomijalne matrice te vidjeli kako ju dobiti, kao i neke poveznice između polinomijalne matrice i njezine Frobeniusove kanonske forme.

Abstract (english)

In this thesis we described some basic terms, theorems and results about polynomial matrices. First of all, we had to define some terms which we had already known like polynomials itself, matrices and some basic operations on polynomials and matrices and after that, we gave the definition of the polynomial matrix. We described the sum of two polynomial matrices, the product of two polynomial matrices and the product of a polynomial matrix and a scalar. After those basic definitions we proved some important theorems about quotients and remainders and also gave some examples to confirm them. There is also an important theorem called Bézout’s theorem which we had generalized. We described linear independence of polynomial vectors and basis for them and also defined the rank of a polynomial matrix. We have noticed that those definitions are quite similar to those of ’classic’ matrices. We showed an example where were given the polynomial vectors which were linearly independent and in the other example they were linearly dependent. We also gave the definition of Smith canonical form of polynomial matrices and the definition of invariant polynomials which we had also used in other chapters. We gave definitions of equivalent matrices and explained the connection between similarity of classical matrices and their polynomial matrices of the first degree. Approaching to the end of the thesis, we defined zeros and elementary divisors of polynomial matrices and we saw that all these definitions are connected with the invariant polynomials. In the end, we defined Frobenius canonical form of a polynomial matrix and also saw some links between a polynomial matrix and its Frobenius canonical form.

• Keywords: polinomi; matrice; Bézoutov teorem; Smithova kanonska forma; invarijantni polinom; Frobeniusova kanonska forma
• Keywords (english): polynomials; matrices; Bézout’s theorem; Smith canonical form; invariant polynomials; Frobenius canonical form
• Language: croatian
• URN:NBN: urn:nbn:hr:217:026131
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-05-08 12:11:59