# A Journey Through Discrete MathematicsGershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

A Journey Through Discrete Mathematics: Gershgorin Disks for Multiple Eigenvalues of Non-negative... [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Journey Through Discrete MathematicsGershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

Editors: Loebl, Martin; Nešetřil, Jaroslav; Thomas, Robin
Springer Journals — May 9, 2017
10 pages

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Publisher
Springer International Publishing
© Springer International publishing AG 2017
ISBN
978-3-319-44478-9
Pages
123 –133
DOI
10.1007/978-3-319-44479-6_6
Publisher site
See Chapter on Publisher Site

### 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.]

Published: May 9, 2017

Keywords: Gershgorin Disks; Eigenvalue Multiplicity; Rearrangement Inequality; Hermitian Positive Semidefinite Matrix; Hesse Configuration