Richard S. Varga, Geršgorin-type eigenvalue inclusion theorems and their sharpness, ETNA (Electronic Transactions on Numerical Analysis), 12(2001), 113-133. MR 2002c:15032. Zbl. 0979.15015.
Dedicated to Professor John Todd, on the occasion of his 90th birthday, May 16, 2001.
Abstract. Here, we investigate the relationships between G( A ), the union of Gerˇsgorin disks, K( A ), the union of Brauer ovals of Cassini, and B( A ), the union of Brualdi lemniscate sets, for eigenvalue inclusions of an n x n complex matrix A . If σ( A ) denotes the spectrum of A , we show here that ˙
( A ) B( A ) K( A ) G( A )
is valid for any weakly irreducible n x n complex matrix A with n x 2.
Further, it is evident that B( A ) can contain the spectra of related n n
matrices. We show here that the spectra of these related matrices can fill out
B( A ).
Finally, if G R ( A ) denotes the minimal Gerˇ sgorin set for A ,
we show that
G R ( A ) B( A ) :