Brauer's Theorem on the Ovals of Cassini
We prove Brauer's Theorem, which is a refinement of the Gershgorin circle Theorem. Brauer's Theorem states that the eigenvalues of a matrix are contained in a union of domains in the complex plane called Ovals of Cassini. These shapes resemble dumbbells. Further generalizations of this type of result fail to hold for matrices in general making Brauer's Theorem one of the sharpest results for bounding the region in which eigenvalues reside.
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