A simplified Brauer’s theorem on matrix eigenvalues

Let A= (aij)∈ Cn×n and ri = Σj≠i. |aij|. Suppose that for each row of A there is at least one nonzero off-diagonal entry. It is proved that all eigenvalues of A are contained in Ω= Uaij≠0,i#j,{z∈C: |z-aii | |z-ajj|≼rirj}. The result reduces the number of ovals in original Brauer's theorem in many cases. Eigenvalues (and associated eigenvectors) that locate in the boundary of Ω are discussed. © 2000, Springer Verlag. All rights reserved.