Math 6610 • Analysis of Numerical Methods I

{The Gershgorin Theorem} Let \(\lambda\) be an eigenvalue of
\(A\in A^{n\times n}\). Show that there exists an
\[
  i\in\{1,2,\cdots,n\} \tag{2}
\]
such that
\[
  |a_{ii}-\lambda|\leq
  \sum_{\substack{j=1\\ i\neq j}}^n |a_{ij}| \tag{3}
\]

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For every eigenvalue \(\lambda\), the inequality (3) describes a circle in the
complex plane called a Gershgorin circle. Let \(S\) be a set that is the union of
\(k\leq n\) Gershgorin circles such that the intersection of \(S\) with all other
Gershgorin circles is empty. Show that \(S\) contains precisely \(k\) eigenvalues
of \(A\) (counting multiplicities). Without proof or counterproof state whether
it is possible for a Gershgorin Circle not to contain any eigenvalue at all.