Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem

l Introduction, The main purpose of this paper is to give generalizations of the well known theorem of Gerschgorin on inclusion or exclusion regions for the eigenvalues of an arbitrary square matrix A. Basically, such exclusion regions arise naturally from results which establish the nonsingularity of A. For example, if A = D + C where D is a nonsingular diagonal matrix, then Householder [7] shows that II-D^CH < 1 in some matrix norm is sufficient to conclude that A is nonsingular. Hence, the set of all complex numbers z for which