A simplified Brauer’s theorem on matrix eigenvalues

Abstract

Let A=(a _ij)∈C^n×n and $r_i = \sum\nolimits_{j \ne i} {\left| {a_{ij} } \right|}$. 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 $\tilde \Omega = \cup _{a_{ij} \ne 0,i \ne j} \left\{ {z \in C:\left| {z--a_{ii} } \right|\left| {z--a_{jj} } \right| \leqslant r_i r_j } \right\}$. 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 $\tilde \Omega$ are discussed.