20.500.12556/DKUM-46906 Popolnoma pozitivne matrike Completely positive matrices Glavna tema magistrske naloge so popolnoma pozitivne matrike, ki so posebni primer pozitivno semidefinitnih matrik. Vsaka realna pozitivno semidefinitna matrika A se lahko zapiše kot A=BB^T, kjer je B realna matrika. V primeru, da je B nenegativna matrika, je matrika A popolnoma pozitivna. Na začetku predstavimo osnovne pojme in definicije realnih matrik, s poudarkom na pozitivno semidefinitnih matrikah. Podamo nekaj primerov in dokažemo osnovne lastnosti teh matrik. V nadaljevanju obravnavamo popolnoma pozitivne matrike. Definiramo Hadamardov in Kroneckerjev produkt ter dokažemo, da sta oba produkta popolnoma pozitivnih matrik popolnoma pozitivni matriki. Spoznamo eno izmed metod, s katero pokažemo, da je dvojno nenegativna matrika popolnoma pozitivna. Definiramo pojem konveksni stožec ter dokažemo, da je množica popolnoma pozitivnih matrik zaprt konveksni stožec. Na algebraični in geometrijski način dokažemo, da so t.i. majhne matrike popolnoma pozitivna. Nazadnje obravnavamo diagonalno dominantne matrike ter dokažemo, da so nenegativne simetrične diagonalno dominantne matrike popolnoma pozitivne. Prav tako definiramo primerjalno matriko in dokažemo, da je matrika A popolnoma pozitivna, če je simetrična nenegativna matrika ter je njena primerjalna matrika pozitivno semidefinitna. The main topic of the master thesis are completely positive matrices, which are the special case of a positive semidefinite matrix. Every real positive semidefinite matrix A can be written in the form A=BB^T, where B is a real matrix. In the case of a nonnegative matrix B the matrix A is completely positive. The first chapter includes some basic terms and definitions of specific real matrices with an emphasis on positive semidefinite matrices. We present some examples and prove the basic properties of these matrices. In the next chapter we consider completely positive matrices. We define Hadamard and Kronecker product and show that both of these products of completely positive matrices are completely positive. We introduce one method which enables us to verify whether the doubly nonnegative matrix is completely positive. We define the concept of a convex cone and show that the set of all completely positive matrices is a closed convex cone. With algebraic and geometric approach we show that small matrices are completely positive. At the end of the thesis we treat diagonally dominant matrices and show that nonnegative symmetric diagonally dominant matrices are completely positive. We also define a comparison matrix and show that the matrix A is completely positive if it is a symmetric nonnegative matrix and if its comparison matrix is positive semidefinite. pozitivno semidefinitna matrika popolnoma pozitivna matrika Hadamardov produkt Kroneckerjev produkt konveksni stožec diagonalno dominantna matrika primerjalna matrika. positive semidefinite matrix completely positive matrix Hadamard product Kronecker product convex cone diagonally dominant matrix comparison matrix. true true false [T. Lešnik] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2014-11-25 18:52:41 2014-12-11 13:56:05 2022-05-19 03:06:37 2021-05-01 03:18:26 2014 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512.64(043.2) 20990984 URN:SI:UM:DK:SYKPNILG MAG_Lesnik_Tina_2014.pdf MAG_Lesnik_Tina_2014.pdf 1 6AD736CB35479A9B1BE6D300C454473A 90cfce751c2e1e7d3f5162efdc7b0c9d85df05e0265c94f110970b79dc2145ca 7150054c-7c0b-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=68985 Fakulteta za naravoslovje in matematiko 0 0 0