Arabian Journal of Mathematics

, Volume 7, Issue 2, pp 91–99 | Cite as

On formulae for the determinant of symmetric pentadiagonal Toeplitz matrices

  • Mohamed Elouafi
Open Access


We show that the characteristic polynomial of a symmetric pentadiagonal Toeplitz matrix is the product of two polynomials given explicitly in terms of the Chebyshev polynomials.

Mathematics Subject Classification

15B05 65F40 33C45 



  1. 1.
    Barrera, M.; Grudsky, S.M.: Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. Oper. Theory Adv. Appl. 259, 179–212 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chu, M.T.; Diele, F.; Ragnion, S.: On the inverse problem of constructing symmetric pentadiagonal Toeplitz matrices from three largest eigenvalues. Inverse Probl. 21, 1879–1894 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Doman, B.G.S.: The Classical Orthogonal Polynomials. World Scientific Publishing Company, Singapore (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Elouafi, M.: A widom like formula for some Toeplitz plus Hankel determinants. J. Math. Anal. Appl. 422(1), 240–249 (2015)Google Scholar
  5. 5.
    Elouafi, M.: An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices. Linear Algebra Appl. 435, 2986–2998 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Elouafi, M.: A note for an explicit formula for the determinant of pentadiagonal and heptadiagonal symmetric Toeplitz matrices. Appl. Math. Comput. 219(9), 4789–4791 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Elouafi, M.: On a relationship between Chebyshev polynomials and Toeplitz determinants. Appl. Math. Comput. 229(25), 27–33 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fasino, D.: Spectral and structural properties of some pentadiagonal symmetric matrices. Calcolo 25, 301–310 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grenander, U.; Szeg ö, G.: Toeplitz Forms and their Applications. Chelsea, New York (1984)Google Scholar
  10. 10.
    Jia, J.T.; Yang, B.T.; Li, S.M.: On a homogeneous recurrence relation for the determinants of general pentadiagonal Toeplitz matrices. Comput. Math. Appl. 71, 1036–1044 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mason, J.C.; Handscomb, D.: Chebyshev Polynomials. Chapman & Hall, New York (2003)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Classes Préparatoites aux Grandes Ecoles d’IngénieursTangierMorocco

Personalised recommendations