A simple generalization of Geršgorin’s theorem

Abstract

It is well known that the spectrum of a given matrix A belongs to the Geršgorin set Γ(A), as well as to the Geršgorin set applied to the transpose of A, Γ(A ^T). So, the spectrum belongs to their intersection. But, if we first intersect i-th Geršgorin disk Γ _i (A) with the corresponding disk $\Gamma_i(A^T)$, and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set Γ(A) ∩ Γ(A ^T). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.