Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / P / 4 / b

The second part of the Gershgorin circle theorem states that if a union of $k$ of the discs is disjoint from every remaining disc, that union contains exactly $k$ eigenvalues, counted with algebraic multiplicity. In particular, a single isolated disc contains precisely one eigenvalue. Connected components of the full union therefore carry the corresponding disc counts.