Invertibility of irreducible matrices (Q1362624)

This paper is in part survey and in part research. The topics are invertibility conditions for irreducible matrices. Classical results such as Gershgorin's circles, Cassini's ovals, Brualdi's cycle theorem, Taussky's relaxation theorem and others are analyzed and extended to cover multibounds. There are new results on eigenvalues and corresponding eigenvectors on the boundary of several of the above spectral inclusion regions, as well as extensions of \textit{V. N. Solovev}'s generalization of Gershgorin's theorem [Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 6, 1285-1302 (1983; Zbl 0551.65017)] on the invertibility of matrices.