Composite matrix inverses and generalized Gershgorin sets

Abstract

In computational matrix algebra, Gershgorin's Theorem [8] and Ostrowski's Theorem[14] play key roles in establishing regularity conditions for complexn×nmatrices. As a result of this importance as well as possible applications in other areas, several extensions of these results have been made (Kotelyanski[9], Ky Fan [5], Brauer[2], Nwokah[10]), for point set matrices. More recently, further extensions have been made to include regularity conditions for composite matrices (matrices partitioned into blocks) (Brenner [3], Van der Sluis[16], Feingold and Varga[6], Cook [4]). Cook's result was motivated by engineering applications but is rather restrictive since, like Feingold and Varga, it requires that the matrices under consideration be diagonally dominant. In this paper, the diagonal dominance condition is replaced by a more general condition (theH-matrix condition). Estimates for the diagonal blocks of block-partitioned matrix are then obtained. The results on inclusion regions for eigenvalues unifiex the previous results of Feingolg and Varga [6], kotelyanski [9], Brauer [2], and generalizes the earlier results of Nwokah [10], from point matrices to block-partitioned matrices. In a sense, these results complement those of brenner [3] Van der Sluis [16], but like those of Cook [4], have a more immediate direct application to control engineering in the area of computer-aided design of composite feedback control systems (Nwokah [12]), which will be briefly reviewed towards the end of the paper.