403 – Some references on Geršgorin’s theorem

I am posting here some references on Geršgorin’s theorem.

• Chapter 1 of Richard S. Varga’s book, Geršgorin and his circles, Springer series in computational mathematics, Springer, 2004.
• Rachid Marsli, and Frank J. Hall. Geometric multiplicities and Geršgorin discs. Amer. Math. Monthly, 120 (5), (2013), 452–455. MR3035444.

(Marsli and Hall have published recently a series of papers further exploring extensions of the theorem. Varga’s book is highly recommended.) If you want to practice some of the topics we have been covered, work through some of the exercises in the posted chapter, and turn them in for some extra credit, by April 8.

Related, though of a different nature, is the following. Geršgorin’s theorem is discussed in section 6:

• Olga Taussky. How I became a torchbearer for matrix theory. Amer. Math. Monthly, 95 (9), (1988), 801–812. MR0967341 (90d:01077).

Here I briefly review the result:

Theorem (Geršgorin, 1931). Let be a complex-valued matrix. For let . If is an eigenvalue of , then for at least one .

To prove this, let be an eigenvector of with eigenvalue , say , and let be such that . Since , we have that , so . From this, the triangle inequality gives us that

and the result follows since .

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