403 – Some references on Geršgorin’s theorem

March 24, 2014

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

(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:

Here I briefly review the result:

Theorem (Geršgorin, 1931). Let A=(a_{ij})_{i,j=1}^n be a complex-valued matrix. For 1\le i\le n let r_i(A)=\sum_{j\ne i}|a_{ij}|. If \lambda is an eigenvalue of A, then {}|\lambda-a_{ii}|\le r_i(A) for at least one i.

To prove this, let x be an eigenvector of A with eigenvalue \lambda, say x=(x_1,\dots,x_n)^T, and let i be such that |x_i|=\max\{|x_k|\mid 1\le k\le n\}. Since Ax=\lambda x, we have that \sum_j a_{ij}x_j=\lambda x_i, so (\lambda-a_{ii})x_i=\sum_{j\ne i}a_{ij}x_j. From this, the triangle inequality gives us that

|\lambda-a_{ii}||x_i|\le\sum_{j\ne i}|a_ij||x_j|\le\sum_{j\ne i}|a_{ij}||x_i|,

and the result follows since x_i\ne0.