H-matrix theory vs. eigenvalue localization

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 Gersgorin 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 Gersgorin-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 Gersgorin set (defined in [19]) is equivalent to the statement that every H-matrix is nonsingular. Since minimal Gersgorin set remained unattainable, a lot of different Gersgorin-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.