| Abstract | U ovom radu osnovni objekti su pozitivne i nenegativne matrice, odnosno one sa pozitivnim i nenegativnim elementima. Središnji pojmovi su svojstvene vrijednosti i njihovi pripadni svojstveni vektori. Među svim svojstvenim vrijednostima najbitnija nam je ona najveća po modulu pa smo time uveli pojam spektralnog radijusa. Za njega smo dokazali mnoge tvrdnje, nejednakosti i formule kako ga možemo izračunati. U prvom poglavlju smo, nakon definiranja osnovnih pojmova, detaljno i postepeno prošli kroz dokaz teorema. Ono je podijeljeno na tri dijela. Baš onako kako se povijesno proširivao teorem, tako i u ovom radu prvo se bavimo sa striktno pozitivnim matricama, a tek onda nakon toga s nenegativnim primitivnim matricama i naposlijetku s nenegativnim ireducibilnim matricama. Pritom uvodimo vezu matrica s teorijom grafova i navodimo karakterizacije ireducibilnosti. Drugo poglavlje ovoga rada je posvećeno raznim primjenama. Prvo je obrađen Fanov teorem koji daje moćan alat u lociranju svojstvenih vrijednosti. On je posebno koristan kod kompliciranih i velikih matrica te tako i Fanov teorem sam po sebi ima mnoštvo primjena. Nakon toga smo naveli nekoliko načina modeliranja rangiranja sportskih timova i pritom nam je Perron-Frobeniusov teorem jamčio da postoje rješenja naših modela i da su ona jedinstvena. Obrađene su i primjene u Markovljevim lancima te objašnjeno kako funkcionira Googleov PageRank algoritam. Naposlijetku, navedena je i primjena na M-matricama kod ekonomskog modela. |
| Abstract (english) | In this thesis, the basic objects are positive and non-negative matrices, i.e. matrices with positive and non-negative elements respectively. Central notions are eigenvalues and their corresponding eigenvectors. Among all eigenvalues, the most important to us is the one largest in modulus, so we introduced the concept of spectral radius. We have proved many claims, inequalities, and formulas for calculating it. In the first chapter, after defining the basic concepts, we went through the proof of the theorem gradually and in detail. It is divided into three parts. Just as the theorem has historically evolved, so in this paper we first deal with strictly positive matrices, and only then with nonnegative primitive matrices and finally with nonnegative irreducible matrices. In doing so, we introduce a connection between matrices and graph theory and state the characterizations of irreducibility. The second chapter of this paper is devoted to various applications. First, Fan’s theorem is given, which provides a powerful tool in locating eigenvalues. It is especially useful for complicated and large matrices, and so the Fan theorem in itself has many applications. After that, we listed several ways of modeling the ranking of sports teams, and the Perron-Frobenius theorem guaranteed that there are solutions for these models and that they are unique. The applications to Markov chains are also discussed, and it is explained how Google’s PageRank algorithm works. Finally, an application to M-matrices in the economic model is also mentioned. |
