On Matrices with Only One Non-SDD Row

Abstract

The class of H-matrices, also known as Generalized Diagonally Dominant (GDD) matrices, plays an important role in many areas of applied linear algebra, as well as in a wide range of applications, such as in dynamical analysis of complex networks that arise in ecology, epidemiology, infectology, neurology, engineering, economy, opinion dynamics, and many other fields. To conclude that the particular dynamical system is (locally) stable, it is sufficient to prove that the corresponding (Jacobian) matrix is an H-matrix with negative diagonal entries. In practice, however, it is very difficult to determine whether a matrix is a non-singular H-matrix or not, so it is valuable to investigate subclasses of H-matrices which are defined by relatively simple and practical criteria. Many subclasses of H-matrices have recently been discussed in detail demonstrating the many benefits they can provide, though one particular subclass has not been fully exploited until now. The aim of this paper is to attract attention to this class and discuss its relation with other more investigated classes, while showing its main advantage, based on its simplicity and elegance. This new approach, which we are presenting in this paper, will be compared with the existing ones, in three possible areas of applications, spectrum localization; maximum norm estimation of the inverse matrix in the point, as well as the block case; and error estimation for LCP problems. The main conclusion is that the importance of our approach grows with the matrix dimension.