20.500.12556/RUL-173511 Obstoj kanonične forme za hkratno podobnost delo diplomskega seminarja Existence of a canonical forms for simultaneous similarity V tem delo obravnavamo klasifikacijo parov kvadratnih matrik $(A, B)$ do hkratne podobnosti natančno, s poudarkom na razredu parov nilpotentnih matrik, ki med seboj komutirajo. Predstavimo Jordanovo in Weyrovo kanonično obliko in množico matrik imenovano reducirana matrična algebra, ki z njo komutira. Pogledamo si Belitskijev algoritem, ki omogoča konstrukcijo predstavnikov razredov hkratne podobnosti glede na dano reducirano matrično algebro, ter tako opišemo postopek kako bi za dva para matrik preverili hkratno podobnost. Posebej analiziramo pare komutirajočih nilpotentnih matrik. Za pare, kjer je prva matrika sestavljena iz dveh Jordanovih blokov različnh velikosti, lahko eksplicitno določimo kanonične predstavnike. Teoretične rezultate podkrepimo z izračuni in primeri, ter si na koncu pogledamo množico predstavnikov za $4\times4$ nilpotentne komutirajoče pare. In this work, we study the classification of pairs of square matrices $(A,B)$ up to simultaneous similarity, with a particular focus on the class of commuting nilpotent matrix pairs. We introduce the Jordan and Weyr canonical forms, as well as the set of matrices known as the reduced matrix algebra, which consists of all matrices commuting with the Weyr matrix. We examine Belitskii's algorithm, which constructs canonical representatives of simultaneous similarity classes with respect to a given reduced matrix algebra, and further describe the process of determining whether two matrix pairs are simultaneously similar. Special attention is devoted to pairs of commuting nilpotent matrices. For pairs in which the first matrix consists of two Jordan blocks of different sizes, we can explicitly determine the canonical representatives. The theoretical results are supported with detailed computations and examples. We conclude by describing the full set of representatives for $4\times4$ commuting nilpotent pairs. hkratna podobnost kanonična oblika Weyrova oblika Jordanova oblika nilpotentna matrika komutirajoči pari Belitskijev algoritem reducirana matrična algebra klasifikacija matrik simultaneous similarity canonical form Weyr form Jordan form nilpotent matrix commuting matrices Belitskii's algorithm reduced matrix algebra matrix classification true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2025-09-18 08:15:35 2025-09-18 08:15:37 2025-11-10 03:35:17 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 153906 250113539 20208.pdf 20208.pdf 1 07234195ADF523ACE92C269A21737733 68cde96fc261b55d3edfd51a3cda3a289bbaf7bd9dd4ff58d671d2683e5daef1 b7df0c65-9456-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=218026 Fakulteta za matematiko in fiziko 0 0 0