Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

Imre Bárány and József Solymosi ()
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Imre Bárány: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
József Solymosi: University of British Columbia, Department of Mathematics

A chapter in A Journey Through Discrete Mathematics, 2017, pp 123-133 from Springer

Abstract: 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.

Keywords: Gershgorin Disks; Eigenvalue Multiplicity; Rearrangement Inequality; Hermitian Positive Semidefinite Matrix; Hesse Configuration (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-44479-6_6

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DOI: 10.1007/978-3-319-44479-6_6

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