Question

Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. Every eigenvalue of A lies in at least one of the m circular disks in the complex plane with centers au and radii Σ (41. Moreover, if n of these disks forrn a connected domain that is disjoint from the other m-n disks, then there are precisely n eigenvalues of A within this domain. Prove the first part of Gerschgorin's theorem. (Hint: Let λ be any eigenvalue of A, and the z corresponding eigenvector with largest entry 1.)

Answer 1

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