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A Journey Through Discrete Mathematics pp 123–133Cite as

Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

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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.

Dedicated to the memory of Jiří Matoušek

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Acknowledgements

This research was supported by ERC Advanced Research Grant no 267165 (DISCONV). Imre Bárány is partially supported by Hungarian National Research Grant K 111827. József Solymosi is partially supported by Hungarian National Research Grant NK 104183 and by an NSERC Discovery Grant. We are indebted to three anonymous referees for very useful comments and information that have improved the presentation of this paper.

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364, Budapest, Hungary

    Imre Bárány

  2. Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, UK

    Imre Bárány

  3. Department of Mathematics, University of British Columbia, 1984 Mathematics Road, V6T 1Z2, Vancouver, BC, Canada

    József Solymosi

Authors
  1. Imre Bárány
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  2. József Solymosi
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Correspondence to József Solymosi .

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Editors and Affiliations

  1. Department of Applied Mathematics, Charles University, Praha, Czech Republic

    Martin Loebl

  2. Computer Science Institute of Charles University, Charles University, Praha, Czech Republic

    Jaroslav Nešetřil

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Robin Thomas

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Bárány, I., Solymosi, J. (2017). Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_6

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