David G. Feingold and Richard S. Varga, Block diagonally dominant matrices and generalizations of the Gerschgorin Circle Theorem, Paci c J. Math. 12 (1962), 1241-1250. MR 27, 1458. Zbl. 109, p. 248.

1. 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
eigenbalues of an arbitrary square matrix *A*. Basically, such exlusion
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 *||D ^{-1}C||<1* in some matrix norm
is sufficient to conclude that

*||(zI-D ^{)-1}C||<1*

evidently contains no eigenvalues of *A*. In a like manner,
Fiedler [4] obtains exclusion regions for the eigenvalues of *A* by
establishing the nonsingularity of *A* through comparisons with *M*-matrices.
Our approach, though not fundamentally different, establishes the nonsingularity
of the matrix *A* by the generalization of the simple concept of a
diagonally dominant matrix. But one of our major results (3) is that these new
exclusion regions can give significant improvements over the usual Gerschgorin
circles in providing bounds for the eigenvalues of *A*.