Generalization of the concept of diagonal dominance with applications to matrix D-stability

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D⊂C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D. Moreover, we show that in some cases, diagonal D-dominance implies (D,D)-stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).