Minimal Gerschgorin sets. II

Bernard Werner Levinger and Richard Steven Varga

Vol. 17 (1966), No. 2, 199–210

DOI: 10.2140/pjm.1966.17.199

Abstract

The Gerschgorin Circle Theorem, which yields n disks whose union contains all the eigenvalues of a given n×n matrix A = (a_i,j), applies equally well to any matrix B = (b_i,j) of the set Ω_A of n×n matrices with b_i,i = a_i,i and |b_i,j| = |a_i,j|, 1 ≦ i, j ≦ n. This union of n disks thus bounds the entire spectrum S(Ω_A) of the matrices in Ω_A. The main result of this paper is a precise characterization of S(Ω_A), which can be determined by extensions of the Gerschgorin Circle Theorem based only on the use of positive diagonal similarity transformations, permutation matrices, and their intersections.

Mathematical Subject Classification

Primary: 15.25

Milestones

Received: 19 May 1964

Published: 1 May 1966