On Brauer-Ostrowski and Brualdi sets

Abstract

For the localization of the spectrum of the eigenvalues of a complex square matrix, the classical Gersgorin Theorem was extended by Ostrowski who used the generalized geometric mean of the row and column sums of the matrix. Ostrowski, and Brauer, extended the previous idea by using generalized geometric means of products of two row and column sums. Finally, by using the Graph Theory, Brualdi extended all of the previous ideas further by considering generalized geometric means of products of two or more than two row and column sums. These localization results can also provide classes of nonsingular matrices. Our main aim in this work is to exploit all the above known results and determine intervals for the parameter(s) alpha (alpha(k)'s) involved so that the localization of the spectrum in question as well as the determination of the associated class of nonsingular matrices are possible. (C) 2014 Elsevier Inc. All rights reserved.