20.500.12556/DKUM-13460 SIMETRIČNE MATRIKE SYMMETRIC MATRICES Simetrične matrike z realnimi elementi so hermitske, zato so unitarno podobne realnim diagonalnim matrikam. Vse njihove lastne vrednosti so realne, prirejene Jordanove kanonične forme pa so diagonalne matrike. V diplomskem delu predstavimo odgovor na naravno vprašanje, ali velja kaj podobnega v primeru, ko so elementi simetričnih matrik kompleksna števila, ki niso vsa realna. Osnovno znanje o hermitskih matrikah podamo v prvem delu. Dopolnimo ga s karakterizacijo matrik, ki so podobne svoji adjungirani matriki. Za simetrične kompleksne matrike obstaja Takagijeva faktorizacija, kar dokažemo v drugem delu. Tu podamo tudi odgovor na zgornje vprašanje, ki je nikalen, saj velja, da je vsaka matrika s kompleksnimi elementi podobna kakšni simetrični matriki s kompleksnimi elementi. Delo zaključimo z opisom tistih kompleksnih matrik, ki so unitarno podobne kakšni kompleksni simetrični matriki. Symmetric matrices with real elements are Hermitian, so they are unitarily similar to real diagonal matrices. All their eigenvalues are real and adjusted Jordan canonical forms are diagonal matrices. In Graduation Thesis we give the answer on the following natural question: Is there anything similar in the case, when the elements of symmetric matrices are complex numbers, not all belonging to the set of real numbers. Basic knowledge about Hermitian matrices is given in the first part. We complete it with characterization of the matrices, which are similar to their adjunct matrix. For symmetric complex matrices there exists Takagi's factorization. We prove this in the second part. Here we give the answer to the question above, which is negative, because every matrix with complex elements is similar to some symmetric matrix with complex elements. We finish our work with the description of those complex matrices, which are unitarily similar to any complex symmetric matrix. simetrična matrika hermitska matrika podobnost unitarna podobnost Takagijeva faktorizacija symmetric matrix Hermitian matrix similarity unitary similarity Takagi's factorization true true false [A. Hrašovec] Slovenski jezik Angleški jezik Diplomsko delo 2010-03-18 23:26:40 2010-05-21 14:38:14 2022-04-12 11:12:40 2023-12-16 03:15:15 2010 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 51(043.2) 17611272 URN:SI:UM:DK:4VLZAY5U UNI_Hrasovec_Andreja_2010.pdf UNI_Hrasovec_Andreja_2010.pdf 1 B2D6DB7EF73521CE2B07E28E3842F600 1804d1fe51fac9a8d7e47a1f77b5c0b32eeb1abd4d49475662b5c9c9d6261566 e8f7b814-7c03-11eb-bb7a-00155d0001ca 20.500.12556/dkum/74624fec-d119-480f-a0dd-8d7710f6f1bb https://dk.um.si/Dokument.php?lang=slv&id=13633 Fakulteta za naravoslovje in matematiko 0 0 0