Abstract
The eigenvalue localization problem is very closely related to the \(H\)-matrix theory. The most elegant example of this relation is the equivalence between the Geršgorin theorem and the theorem about nonsingularity of SDD (strictly diagonally dominant) matrices, which is a starting point for further beautiful results in the book of Varga [19]. Furthermore, the corresponding Geršgorin-type theorem is equivalent to the statement that each matrix from a particular subclass of \(H\)-matrices is nonsingular. Finally, the statement that all eigenvalues of a given matrix belong to minimal Geršgorin set (defined in [19]) is equivalent to the statement that every \(H\)-matrix is nonsingular. Since minimal Geršgorin set remained unattainable, a lot of different Geršgorin-type areas for eigenvalues has been developed recently. Along with them, a lot of new subclasses of \(H\)-matrices were obtained. A survey of recent results in both areas, as well as their relationships, will be presented in this paper.
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Communicated by Michael Neumann.
Dedicated to Richard S. Varga.
This work is partly supported by Republic of Serbia, Ministry of Science and Environmental Protection, grant 144025.
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Cvetković, L. H-matrix theory vs. eigenvalue localization. Numer Algor 42, 229–245 (2006). https://doi.org/10.1007/s11075-006-9029-3
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DOI: https://doi.org/10.1007/s11075-006-9029-3