Totalno pozitivne matrice

master's thesis

• Author: Nives Kunjašić
• Mentor: Ljiljana Arambašić (mentor)
• Committee member: Ljiljana Arambašić (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Mario Basletić (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Matko Milin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavili smo se totalno pozitivnim i striktno totalno pozitivnim matricama. U prvom poglavlju uvodimo osnovne pojmove, navodimo primjere te proučavamo operacije koje čuvaju promatrane klase matrica kao i metode kako iz postojećih (striktno) totalno pozitivnih matrica konstruirati nove matrice iz iste klase. Između ostalog, diskutiramo kako elementarne transformacije nad retcima i stupcima matrice utječu na (striktnu) totalnu pozitivnost matrice, dokazujemo da transponiranje čuva (striktnu) totalnu pozitivnost, kako inverz (striktno) totalno pozitivne matrice, koji ne mora biti striktno totalno pozitivna matrica, modificirati do matrice s ovim svojstom i slično. S obzirom da definicija (striktno) totalne pozitivnosti zahtijeva provjeru pozitivnosti odnosno nenegativnosti svih minora zadane matrice, poželjno je naći dovoljne uvjete kojima bi se broj tih provjera smanjio. Neki od rezultata ovog tipa su Feketeova lema te Gasca-Penna teorem. U posljednjem poglavlju naveli smo primjere poznatih matrica koje pripadaju promatranoj klasi matrica. Najpoznatiji primjer je Pascalova matrica, čiji su elementi binomni koeficijenti. Nadalje, Cauchyjeva te Vandermondeva matrica su, uz pogodno izabrane nizove brojeva koji ih definiraju, također (striktno) totalno pozitivne. U posljednjoj sekciji se proučavaju Jacobijeva, Hankelova i Toeplitzova matrica.

Abstract (english)

In this paper we studied totally positive and strictly totally positive matrices. In the first chapter we introduced basic concepts, examples, discussed some operations that preserve the considered classes of matrices as well as the methods for constructing new matrices of the same class from the existing ones. In particular, we discuss how elementary transformations over rows and columns of a matrix effect the (strict) total positivity of matrices, we prove that transposition preserves (strict) total positivity, and show how to modify the inverse of a (strictly) totally positive matrix, which does not have to be a (strictly) totally positive matrix, to a matrix with this property, etc. Since the definition of (strict) total positivitness requires verification of the positivity or nonnegativity of all minors of a given matrix, it is desirable to find sufficient conditions which reduce the number of these verifications. Some of the results of this type are Fekete’s lemma and the Gasca-Penna theorem. In the last chapter we have provided examples of known matrices belonging to the considered class of matrices. The best known example is the Pascal matrix, whose elements are binomial coefficients. Further, the Cauchy and the Vandermonde matrix, with the suitably chosen entries, are also (strictly) totally positive matrices. In the last section, we studied Jacobi, Hankel and Toeplitz matrices.

• Keywords: totalno pozitivne matrice; Feketeova lema; Gasca-Penna teorem; Pascalova matrica; Cauchyjeva matrica; Vandermondeva matrica; Jacobijeva matrica; Hankelova matrica; Toeplitzova matrica
• Keywords (english): totally positive matrices; Fekete’s lemma; Gasca-Penna theorem; Pascal matrix; Cauchy matrix; Vandermonde matrix; Jacobi matrix; Hankel matrix; Toeplitz matrix
• Language: croatian
• URN:NBN: urn:nbn:hr:217:714032
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-30 11:07:59