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Α study cοncerning the pοsitive semi-definite prοperty fοr similarity matrices and fοr dοubly stοchastic matrices with sοme applicatiοns

Rafic Nader 1
1 Equipe CODAG - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen
Abstract : Matrix theory has shown its importance by its wide range of applications in different fields such as statistics, machine learning, economics and signal processing. This thesis concerns three main axis related to two fundamental objects of study in matrix theory and that arise naturally in many applications, that are positive semi-definite matrices and doubly stochastic matrices.One concept which stems naturally from machine learning area and is related to the positive semi-definite property, is the one of similarity matrices. In fact, similarity matrices that are positive semi-definite are of particular importance because of their ability to define metric distances. This thesis will explore the latter desirable structure for a list of similarity matrices found in the literature. Moreover, we present new results concerning the strictly positive definite and the three positive semi-definite properties of particular similarity matrices. A detailed discussion of the many applications of all these properties in various fields is also established.On the other hand, an interesting research field in matrix analysis involves the study of roots of stochastic matrices which is important in Markov chain models in finance and healthcare. We extend the analysis of this problem to positive semi-definite doubly stochastic matrices.Our contributions include some geometrical properties of the set of all positive semi-definite doubly stochastic matrices of order n with nonnegative pth roots for a given integer p. We also present methods for finding classes of positive semi-definite doubly stochastic matrices that have doubly stochastic pth roots for all p, by making use of the theory of M-Matrices and the symmetric doubly stochastic inverse eigenvalue problem (SDIEP), which is also of independent interest.In the context of the SDIEP, which is the problem of characterising those lists of real numbers which are realisable as the spectrum of some symmetric doubly stochastic matrix, we present some new results along this line. In particular, we propose to use a recursive method on constructing doubly stochastic matrices from smaller size matrices with known spectra to obtain new independent sufficient conditions for SDIEP. Finally, we focus our attention on the realizability by a symmetric doubly stochastic matrix of normalised Suleimanova spectra which is a normalized variant of the spectra introduced by Suleimanova. In particular, we prove that such spectra is not always realizable for odd orders and we construct three families of sufficient conditions that make a refinement for previously known sufficient conditions for SDIEP in the particular case of normalized Suleimanova spectra.
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Submitted on : Friday, December 18, 2020 - 11:47:19 AM
Last modification on : Wednesday, November 3, 2021 - 5:07:49 AM
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Rafic Nader. Α study cοncerning the pοsitive semi-definite prοperty fοr similarity matrices and fοr dοubly stοchastic matrices with sοme applicatiοns. Computer Science [cs]. Université de Caen (France), 2019. English. ⟨tel-03081681⟩



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