A simple generalization of Gergorin's theorem

作者：Cvetkovic Ljiljana; Kostic Vladimir; Bru Rafael; Pedroche Francisco^*

来源：Advances in Computational Mathematics, 2011, 35(2-4): 271-280.

DOI：10.1007/s10444-009-9143-6

• Eigenvalue localization
• H-matrices
• alpha-matrices

It is well known that the spectrum of a given matrix A belongs to the Gersgorin set Gamma (A), as well as to the Gersgorin set applied to the transpose of A, Gamma(A(T)). So, the spectrum belongs to their intersection. But, if we first intersect i-th Gersgorin disk Gamma(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 Gamma (A) boolean AND Gamma (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.

• 出版日期2011-11