Journal of Mathematical Chemistry

, Volume 56, Issue 4, pp 1234–1249 | Cite as

Explicit form of determinants and inverse matrices of Tribonacci r-circulant type matrices

  • Xiaoyu Jiang
  • Kicheon HongEmail author
Original Paper


The determinants and inverses of Tribonacci r-circulant type matrices are discussed in the paper. Firstly, Tribonacci r-circulant type matrices are defined. In addition, we show the invertibility of the Tribonacci r-circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between r-circulant and r-left circulant, the invertibility of the Tribonacci r-left circulant matrix are also discussed. Finally, the determinants and the inverse matrices of the these matrices are given, respectively.


r-Circulant matrix r-Left circulant matrix Determinant Inverse Tribonacci number 

Mathematics Subject Classification

15A09 15A15 15B05 65F40 11B39 



This work was supported by the GRRC program of Gyeonggi Province [(GRRC SUWON 2016-B3), Development of cloud Computing-based Intelligent Video Security Surveillance System with Active Tracking Technology]. Their support is gratefully acknowledged.


  1. 1.
    S. Arimoto, Repeat space theory applied to carbon nanotubes and related molecular networks. J. Math. Chem. 41(3), 231–269 (2007)CrossRefGoogle Scholar
  2. 2.
    L.Z. Zhang, S.L. Wei, F.L. Lu, The number of Kekul\(\acute{e}\) structures of polyominos on the torus. J. Math. Chem. 51, 354–368 (2013)CrossRefGoogle Scholar
  3. 3.
    F.l Lu, Y.J. Gong, H.C. Zhou, The spectrum and spanning trees of polyominos on the torus. J. Math. Chem 52, 1841–1847 (2014)CrossRefGoogle Scholar
  4. 4.
    Q.X. Kong, J.T Jia, A structure-preserving algorithm for linear systems with circulant pentadiagonal coefficient matrices. J. Math. Chem. 53, 1617–1633 (2015)CrossRefGoogle Scholar
  5. 5.
    M.V. Houteghem, T. Verstraelen, D.V. Neck, C. Kirschhock, J.A. Martens, M. Waroquier, V.V. Speybroeck, Atomic velocity projection method: a new analysis method for vibrational spectra in terms of internal coordinates for a better understanding of zeolite nanogrowth. J. Chem. Theory Comput. 7, 1045–1061 (2011)CrossRefGoogle Scholar
  6. 6.
    K. Balasubramanian, Computational strategies for the generation of equivalence classes of hadamard matrices. J. Chem. Inf. Comput. Sci. 35, 581–589 (1995)CrossRefGoogle Scholar
  7. 7.
    A.T. Wood, G. Chan, Simulation of stationary Gaussian processes in [0,1]\(^{d}\). J. Comput. Graph. Stat. 3, 409–432 (1994)Google Scholar
  8. 8.
    W. Min, B.P. English, G.B. Luo, B.J. Cherayil, S.C. Kou, X.S. Xie, Fluctuating enzymes: lessons from single-molecule studies. Acc. Chem. Res. 38, 923–931 (2005)CrossRefGoogle Scholar
  9. 9.
    D. Yerchuck, A. Dovlatova, Quantum optics effects in quasi-one-dimensional and two-dimensional carbon materials. J. Phys. Chem. 116, 63–80 (2012)CrossRefGoogle Scholar
  10. 10.
    P.J. Davis, Circulant Matrices (Wiley, New York, 1979)Google Scholar
  11. 11.
    Z.L. Jiang, Z.X. Zhou, Circulant Matrices (Chengdu Technology University Publishing Company, Chengdu, 1999)Google Scholar
  12. 12.
    Z.L. Jiang, T.T. Xu, Norm estimates of \(\omega \)-circulant operator matrices and isomorphic operators for \(\omega \)-circulant algebra. Sci. China Math. 59(2), 351–366 (2016)CrossRefGoogle Scholar
  13. 13.
    Z.L. Jiang, Y.C. Qiao, S.D. Wang, Norm equalities and inequalities for three circulant operator matrices. Acta Math. Sin. Engl. Ser. (2016). Google Scholar
  14. 14.
    J.L. Jiang, H.X. Xin, H.W. Wang, On computing of positive integer powers for \(r\)-circulant matrices. Appl. Math. Comput. 265, 409–413 (2015)CrossRefGoogle Scholar
  15. 15.
    M.J. Narasimha, Linear convolution using skew-cyclic convolutions. IEEE Signal Process. Lett. 14, 173–176 (2007)CrossRefGoogle Scholar
  16. 16.
    D. Bertaccini, M.K. Ng, Block \({\omega }\)-circulant preconditioners for the systems of differential equations. Calcolo 40, 71–90 (2003)Google Scholar
  17. 17.
    L.D. Zheng, Fibonacci–Lucas quasi-cyclic matrices. Fibonacci Q. 40, 280–286 (2002)Google Scholar
  18. 18.
    S.Q. Shen, J.M. Cen, On the bounds for the norms of \(r\)-circulant matrices with the Fibonacci and Lucas numbers. Appl. Math. Comput. 216, 2891–2897 (2010)CrossRefGoogle Scholar
  19. 19.
    Z.L. Jiang, J.W. Zhou, A note on spectral norms of even-order \(r\)-circulant matrices. Appl. Math. Comput. 250, 368–371 (2015)Google Scholar
  20. 20.
    E.C. Boman, The Moore–Penrose pseudoinverse of an arbitrary, square, \(k\)-circulant matrix. Linear Multilinear Algeb. 50, 175–179 (2010)CrossRefGoogle Scholar
  21. 21.
    Z.L. Jiang, Nonsingularity for two circulant type matrices. J. Math. Pract. Theory. 2, 52–58 (1995)Google Scholar
  22. 22.
    Y. Mei, Computing the square roots of a class of circulant matrices. J. Appl. Math. Article ID 647623, 15 (2012)Google Scholar
  23. 23.
    J. Li, Z. L. Jiang, F. L. Lu, Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers. Abstr. Appl. Anal. Article ID 381829, 9 (2014)Google Scholar
  24. 24.
    Z. L. Jiang , Y. P. Gong, Y. Gao, Circulant type matrices with the sum and product of Fibonacci and Lucas numbers. Abstr. Appl. Anal. Article ID 375251, 12 (2014)Google Scholar
  25. 25.
    Z. L Jiang, Y. P. Gong, Y. Gao, Invertibility and explicit inverses of circulant-type matrices with \(k\)-Fibonacci and \(k\)-Lucas numbers. Abstr. Appl. Anal. Article ID 238953, 10 (2014)Google Scholar
  26. 26.
    X. Y. Jiang, K. C. Hong, Exact determinants of some special circulant matrices involving four kinds of famous numbers. Abstr. Appl. Anal. Article ID 273680, 12 (2014)Google Scholar
  27. 27.
    D. Bozkurt, T.Y. Tam, Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas numbers. Appl. Math. Comput. 219, 544–551 (2012)CrossRefGoogle Scholar
  28. 28.
    S.Q. Shen, J.M. Cen, Y. Hao, On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers. Appl. Math. Comput. 21, 9790–9797 (2011)Google Scholar
  29. 29.
    L. Liu and Z. L. Jiang, Explicit form of the inverse matrices of Tribonacci circulant type matrices. Abstr. Appl. Anal. Article ID 169726, 10 (2015)Google Scholar
  30. 30.
    X.Y. Jiang, K.C. Hong, Explicit inverse matrices of Tribonacci skew circulant type matrices. Appl. Math. Comput. 268, 93–102 (2015)CrossRefGoogle Scholar
  31. 31.
    Y.P. Zheng, S. Shon, Exact determinants and inverses of generalized Lucas skew circulant type matrices. Appl. Math. Comput. 270, 105–113 (2015)CrossRefGoogle Scholar
  32. 32.
    M. Elia, Derived sequences, the Tribonacci recurrence and cubic forms. Fibonacci Q. 39, 107–115 (2001)Google Scholar
  33. 33.
    B. Balof, Restricted tiling and bijections. J. Integer Seq. 15(2), Article 12.2.3 (2012)Google Scholar
  34. 34.
    S. Rabinowitz, Algorithmic manipulation of third-order linear recurrences. Fibonacci Q. 34, 447–464 (1996)Google Scholar
  35. 35.
    J.W. Zhou, Z.L. Jiang, The spectral norms of \(g\)-circulant matrices with classical Fibonacci and Lucas numbers entries. Appl. Math. Comput. 233, 582–587 (2014)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Information and Telecommunications EngineeringThe University of SuwonHwaseong-siKorea
  2. 2.College of InformaticsLinyi UniversityLinyiChina

Personalised recommendations