Some new inequalities on geometric multiplicities and Gersgorin discs

Abstract

In this paper the authors continue their work on geometric multiplicities and Geršgorin discs done in a series of four recent papers. The new results involve principal submatrices, an upper bound on the absolute value of an eigenvalue, the rank of a matrix, non-real eigenvalues, and powers of matrices. Some consequences of the results 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.…”

Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

“…An interesting and recent theorem of Marsli and Hall [5] states that if an eigenvalue of a matrix A has geometric multiplicity k, then it lies in at least k of the Gershgorin disks of A. They have extended this result in subsequent papers [3,6,7,8]. Here we focus on the k = 2 case for non-negative matrices.…”