A refinement of Gerschgorin’s circle theorem
Recall the basic Gerschgorin circle theorem:
Let
and define
. Define the Gerschgorin discs
, then
.
Here’s a useful enhancement to the theorem: if the union of the discs can be split into two disjoint regions, the first consisting of the union of 
discs and the second consisting of the union of 
discs, then there are eigenvalues in the first region and in the second.
Prove it! Hint: use the continuity of eigenvalues as you vary the matrix.