A counterexample connected with Geršgorin's theorem
Let \(A=(a_{ij})\) be an \(n \times n\) complex matrix. It is known that a union of disks in the complex plane contains all eigenvalues of \(A\). In this classical Gershgorin theorem each disk has its center at a single diagonal element \(a_{ii}\) of \(A\). \textit{A. Brauer}'s generalization [Duke Math. J. 14, 21-26 (1947; Zbl 0029.33701)] of Gershgorin's theorem provides a second order union. In this note the authors exhibit examples demonstrating that a third order Gershgorin region need not contain any eigenvalue.
scientific article; zbMATH DE number 614179