Some refinements of Gersgorin discs

Abstract

This paper refines the work on Geršgorin Discs in the article "Geometric Multiplicities and Geršgorin Discs", The American Mathematical Monthly, 120(2013), 452-455 by R. Marsli and F. Hall. Some consequences of this refinement and examples are provided.

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

“…Varga's nice book [31] surveys various applications and extensions of this important theorem. Recently, Marsli and Hall [25] found an interesting result, which states that if λ is an eigenvalue of an n × n matrix A with geometric multiplicity k, then λ is in at least k of the n Gerŝgorin discs of A. Fiedler et al [12] proved that for a triple of positive integers k, r, t with k ≤ r ≤ t, there is a t × t complex matrix A and an eigenvalue λ of A such that λ has geometric multiplicity k and algebraic multiplicity t, and λ is in precisely r Gerŝgorin discs of A. Marsli and Hall extended these results in subsequent papers [24,26,27]. Bárány and Solymosi [6] showed that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk.…”

Some new inequalities on geometric multiplicities and Gersgorin discs

“…In particular, the following result was proved in [3]. More recently, we obtained a refinement of Theorem 1.1 in [5]. Theorem 1.2.…”

“…The next theorem is a generalization of Theorem 3.9 in [5] Theorem 2.7. Let A ∈ M n and let • be a matrix norm.…”

“…A matrix A ∈ M n has n Geršgorin discs D i , some of which may degenerate into points and some of which may be duplicates, as in the trivial example of an identity matrix. Recently, the authors extended the Geršgorin theory in the articles [1], [3], [4], and [5]. In particular, the following result was proved in [3].…”

“…(Corollary 3.2 in [5]) Let A ∈ M n be such that|a ii | = β, i = 1, 2, ..., n |a ij | = α, i = j.If λ is an eigenvalue of A with geometric multiplicity k, then |λ| ≤ α k (C k ) = β + (n − k)α.…”