Benefits from generalized diagonal dominance

Generalized diagonal dominance, a specific property of the matrix, can be very useful in various fields of the applied linear algebra, as a tool for discovering and proving new results. First, it gives possibility to be able to define different subclasses of H-matrices (papers [3], [11]). Second, it suggests a way to deduce and prove new theorems on the eigenvalue localization (papers [2], [4], [5], [10], [14], [15]). Third, it gives the possibility to obtain new results on the convergence of iterative methods (papers [1], [6], [7]). Fourth, it is a tool for proving new properties of Schur complements (papers [8], [12]) and new features of the matrix sums (paper [9]). Finally, the generalized diagonal dominance was proven to be an indispensable tool in getting the first results on the localization of generalized eigenvalues (paper [13]).