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Special Matrices

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On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix

Jorma K. Merikoski / Pentti Haukkanen / Mika Mattila / Timo Tossavainen
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  • Department of Arts, Communication and Education, Lulea University of Technology, SE-97187 Lulea, Sweden
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Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/spma-2018-0003


Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

Keywords: Euclidean norm; Frobenius norm; generalized Fibonacci numbers; r-circulant matrix; spectral norm


  • [1] M. Bahsi, On the norms of r-circulant matrices with the hyperharmonic numbers, J. Math. Inequal. 10, 445-458. (2016)Google Scholar

  • [2] M. Bahsi, S. Solak, On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers, J. Math. Inequal. 8, 693-705. (2014)Google Scholar

  • [3] A. Chandoul, On the norms of r-circulant matrices with generalized Fibonacci numbers. J. Algebra Comb. Discrete Struct. Appl. 4, 13-21. (2017)Google Scholar

  • [4] P. J. Davis, Circulant Matrices. (John Wiley & Sons, New York, 1979)Google Scholar

  • [5] C. He, J. Ma, K. Zhang, Z. Wang, The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers. J. Inequal. Appl. 2015:72. (2015)Google Scholar

  • [6] R. A. Horn, C. R. Johnson, Matrix Analysis, Second Edition. (Cambridge University Press, New York, 2013)Google Scholar

  • [7] E. G. Kocer, T. Mansour, N. Tuglu, Norms of circulant and semicirculant matrices with Horadam’s numbers, Ars Combin. 85, 353-359. (2007)Google Scholar

  • [8] T. Koshy, Fibonacci and Lucas Numbers with Applications. (John Wiley & Sons, New York, 2001)Google Scholar

  • [9] C. Köme, Y. Yazlik, On the spectral norms of r-circulantmatrices with the biperiodic Fibonacci and Lucas numbers. J. Inequal. Appl. 2017:192. (2017)Google Scholar

  • [10] J. K. Merikoski, P. Haukkanen, M.Mattila, T. Tossavainen, The spectral norm of a Horadamcirculantmatrix. arXiv: 1705.03494 [math.NT]Google Scholar

  • [11] Z. Raza, M. A. Ali, On the norms of some special matrices with generalized Fibonacci sequence. J. Appl. Math. Inform. 33, 593-605. (2015)Google Scholar

  • [12] S. Shen, J. Cen, On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers. Appl. Math. Comput. 216, 2891-2897. (2010)Google Scholar

  • [13] Y. Yazlik, N. Taskara, On the norms of an r-circulant matrix with the generalized k-Horadam numbers. J. Inequal. Appl. 2013:394. (2013)Google Scholar

About the article

Received: 2017-10-09

Accepted: 2017-12-26

Published Online: 2018-01-24

Citation Information: Special Matrices, Volume 6, Issue 1, Pages 23–36, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2018-0003.

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© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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