Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

Abstract

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show 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. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper. 1 arXiv:1609.07439v1 [math.CO]

Gershgorin disks for multiple and close eigenvalues

“…Bounding S comes form Theorem 2 in [3] and, in a slightly more general form, from Lemma 2.4 of [8] in the form of the following fact.…”

“…Under some special conditions the Gershgorin bound has recently been improved by Bárány and Solymosi [3] followed by Hall and Marsli [8]. Assume A is an n × n real matrix and let b i1 .…”

“…The following result is implicit in [3]; see the equations after the proof of Theorem 2 there. A detailed proof appears in Hall and Marsli [8].…”

On the Ger\v{s}gorin disks of distance matrices of graphs

“…Marsli and Hall [11] gave an interesting result stating that if M has the eigenvalue λ with algebraic multiplicity k, then λ lies in at least k of the n Geršgorin disks. Bárány and Solymosi [7] showed that if M is a nonnegative real matrix and λ is an eigenvalue of M with geometric multiplicity at least k, then it is in a smaller disk. More literature about the Geršgorin disk theorem can be found in [10,15] and the references therein.…”

“…8,7 ) is the circle (interval) with center at 35 and passing through the origin: [0, 70]. The approximate eigenvalues (precision 10 −4 ) of D Q (P K 8,7 ) are 108.3722, 75.6213, 62.2837, 51.7205, 43.5872, 41(6) , 37.8609, 34.9826, 33.8082, and 19.7637, where the superscript (6) represents the algebraic multiplicity of the eigenvalue 41.…”