On general principles of eigenvalue localisations via diagonal dominance

Abstract

© 2014, Springer Science+Business Media New York. This paper suggests a unifying framework for matrix spectra localizations that originate from different generalizations of strictly diagonally dominant matrices. Although a lot of results of this kind have been published over the years, in many papers same properties were proven for every specific localization area using basically the same techniques. For that reason, here, we introduce a concept of DD-type classes of matrices and show how to construct eigenvalue localization sets. For such sets we then prove some general principles and obtain as corollaries many singular results that occur in the literature. Moreover, obtained principles can be used to construct and use novel Geršgorin-like localization areas. To illustrate this, we first prove a new nonsingularity result and then use established principles to obtain the corresponding localization set and its several properties. In addition, some new results on eigenvalue separation lines and upper bounds for spectral radius are obtained, too.