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A simple derivation of the eigenvalues of a tridiagonal matrix arising in biogeography

Volume 42 / 2015

Qassem M. Al-Hassan, Mowaffaq Hajja Applicationes Mathematicae 42 (2015), 23-27 MSC: Primary 15A18; Secondary 15B99, 92D25, 92D40. DOI: 10.4064/am42-1-3


In investigating a certain optimization problem in biogeography, Simon [IEEE Trans. Evolutionary Comput. 12 (2008), 702–713] encountered a certain specially structured tridiagonal matrix and made a conjecture regarding its eigenvalues. A few years later, the validity of the conjecture was established by Igelnik and Simon [Appl. Math. Comput. 218 (2011), 195–201]. In this paper, we give another proof of this conjecture that is much shorter, almost computation-free, and does not resort to the eigenvectors of the matrix.


  • Qassem M. Al-HassanDepartment of Mathematics
    Yarmouk University
    Irbid, Jordan
  • Mowaffaq HajjaDepartment of Mathematics
    Yarmouk University, Irbid

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