H-matrix theory vs. eigenvalue localization
Tóm tắt
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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