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Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

In: A Journey Through Discrete Mathematics

Author

Listed:
  • Imre Bárány

    (Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
    University College London, Department of Mathematics)

  • József Solymosi

    (University of British Columbia, Department of Mathematics)

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.

Suggested Citation

  • Imre Bárány & József Solymosi, 2017. "Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices," Springer Books, in: Martin Loebl & Jaroslav Nešetřil & Robin Thomas (ed.), A Journey Through Discrete Mathematics, pages 123-133, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-44479-6_6
    DOI: 10.1007/978-3-319-44479-6_6
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