Potence pozitivnih matrik
delo diplomskega seminarja
Abstract
Diplomska naloga se osredotoča na potence pozitivnih matrik. Te so tesno povezane z lastnimi vrednostmi in lastnimi vektorji matrik, zato delo razloži postopek za njihovo računanje in lastnosti matrik v povezavi z njimi. Na podlagi Jordanove kanonične forme je v delu razložen preprostejši postopek za računanje potenc in hkrati tudi drugih funkcij matrik. Razložene so tudi lastnosti potenc stohastičnih oziroma verjetnostnih matrik. Osrednja izreka dela sta Perron-Frobeniusov in Perronov izrek. Prvi razloži lastnosti pozitivnih matrik, ki jih nato drugi uporabi pri računanju limit zaporedja potenc pozitivnih matrik. Vse preučeno je na koncu uporabljeno na primerih iz realnega življenja, kjer lahko vidimo uporabnost potenc pozitivnih matrik in razlog zakaj smo to temo sploh preučevali.
Keywords
matematika;lastne vrednosti;lastni vektorji;pozitivne matrike;nenegativne matrike;Perron-Frobeniusov izrek;Perronov izrek;potence matrik;Jordanova kanonična forma;stohastične matrike;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2021 |
| Typology: | 2.11 - Undergraduate Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [T. Žitko] |
| UDC: | 512 |
| COBISS: |
78748163
|
| Views: | 1336 |
| Downloads: | 98 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
|---|---|
| Secondary title: | Powers of positive matrices |
| Secondary abstract: | The thesis focuses on powers of positive matrices. They are closely related to eigenvalues and eigenvectors of said matrices. For this reason the thesis explains the procedure for calculating eigenpairs and their characteristics. On the basis of Jordan normal form the thesis explains a simpler way of calculating powers of matrices and also other matrix functions. It also explains the characteristics of powers of stochastic or probability matrices. The centre theorems of this thesis are Perron-Frobenius and Perron theorem. The first focuses on characteristics of positive matrices, which then the second uses to compute limits of sequences of powers of positive matrices. Everything we learn is then used in examples from real life, where we can see the usefulness of powers of positive matrices and the reason we started studying them in the first place. |
| Secondary keywords: | mathematics;eigenvalues;eigenvectors;positive matrices;nonnegative matrices;Perron-Frobenius theorem;Perron theorem;powers of matrices;Jordan normal form;stochastic matrices; |
| Type (COBISS): | Final seminar paper |
| Study programme: | 0 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja |
| Pages: | 24 str. |
| ID: | 13539686 |
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