Gershgorin Disks for Multiple Eigenvalues of Non-negative Matrices

  • Imre Bárány
  • József Solymosi


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.



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.


  1. 1.
    A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (SIAM, Philadelphia, 1994)CrossRefMATHGoogle Scholar
  2. 2.
    G.M. Del Corso, Estimating an eigenvector by the power method with a random start. SIAM. J. Matrix Anal. Appl. 18, 913–937 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. Fiedler, F.J. Hall, R. Marsli, Gershgorin discs revisited. J. Linear Algebra Appl. 438, 598–603 (2013)CrossRefMATHGoogle Scholar
  4. 4.
    S. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 6, 749–754 (1931)MATHGoogle Scholar
  5. 5.
    O. Hesse, Über die Wendepunkte der Curven dritter Ordnung. J. Reine Angew. Math. 28, 97–102 (1844)MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Lovász, Steinitz Representations of Polyhedra and the Colin de Verdière Number. J. Combin. Theory B 82, 223–236 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    R. Marsli, F.J. Hall, Geometric multiplicities and Gershgorin discs. Am. Math. Mon. 120, 452–455 (2013)CrossRefMATHGoogle Scholar
  8. 8.
    R. Marsli, F.J. Hall, Some refinements of Gershgorin discs. Int. J. Algebra 7, 573–580 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    R. Marsli, F.J. Hall, Further results on Gershgorin discs. J. Linear Algebra Appl. 439, 189–195 (2013)CrossRefMATHGoogle Scholar
  10. 10.
    R. Marsli, F.J. Hall, Some new inequalities on geometric multiplicities and Gershgorin discs. Int. J. Algebra 8, 135–147 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    P. McMullen, Transforms, diagrams and representations, in Contributions to Geometry. Proceedings of the Geometry-Symposium, Siegen, 1978 (Birkhäuser, Basel/Boston, 1979), pp. 92–130Google Scholar
  12. 12.
    A. Roy, Minimal Euclidean representations of graphs. Discret. Math. 310, 727–733 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    J.M. Steele, The Cauchy-Schwarz Master Class. An Introduction to the Art of Mathematical Inequalities (Cambridge University Press, New York, 2004)Google Scholar
  14. 14.
    H. van der Holst, L. Lovász, A. Schrijver, The Colin de Verdière graph parameter, in Graph Theory and Combinatorial Biology. Bolyai Society Mathematical Studies, vol. 7 (János Bolyai Mathematical Society, Budapest, 1999), pp. 29–85Google Scholar
  15. 15.
    R.S. Varga, Gershgorin and His Circles (Springer, Berlin, 2004)CrossRefMATHGoogle Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of Mathematics, Hungarian Academy of SciencesBudapestHungary
  2. 2.Department of MathematicsUniversity College LondonLondonUK
  3. 3.Department of MathematicsUniversity of British ColumbiaVancouverCanada

Personalised recommendations