Matrične potence

V delu diplomskega seminarja obravnavamo matrične potence in zaporedja matričnih potenc. Sprva navedemo motivacijska zgleda in naredimo pregled pojmov iz linearne algebre, potrebnih za razumevanje besedila. Nato pogledamo matrične polinome in funkcije ter navedemo nekaj osnovnih pojmov iz spektralne teorije. V osrednjem delu si ogledamo koordinatna zaporedja, asimptotsko obnašanje matričnih zaporedij in Cesàrovo konvergenco. Na koncu znanje, pridobljeno tekom diplomskega dela, uporabimo za analizo markovskih verig. In the thesis we discuss matrix powers and sequences of matrix powers. We first give motivating examples and make an overview of the concepts from linear algebra needed to understand the text. We look at matrix polynomials and functions and list some basic concepts from spectral theory. In the central part, we look at coordinate sequences, asymptotic behavior of matrix sequences, and Cesàro summability. Finally, the knowledge gained during the thesis is used to analyze Markov chains.