Abstract and Applied Analysis

Invertibility and Explicit Inverses of Circulant-Type Matrices with k -Fibonacci and k -Lucas Numbers

Zhaolin Jiang, Yanpeng Gong, and Yun Gao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with the k -Fibonacci and k -Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).

Article information

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 238953, 9 pages.

First available in Project Euclid: 3 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Jiang, Zhaolin; Gong, Yanpeng; Gao, Yun. Invertibility and Explicit Inverses of Circulant-Type Matrices with $k$ -Fibonacci and $k$ -Lucas Numbers. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 238953, 9 pages. doi:10.1155/2014/238953. https://projecteuclid.org/euclid.aaa/1412361157

Export citation


  • A. C. Wilde, “Differential equations involving circulant matrices,” The Rocky Mountain Journal of Mathematics, vol. 13, no. 1, pp. 1–13, 1983.
  • B. Voorhees and A. Nip, “Ordinary differential equations with star structure,” Journal of Dynamical Systems and Geometric Theories, vol. 3, no. 2, pp. 121–152, 2005.
  • C. Zhang, H. Chen, and L. Wang, “Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations,” Numerical Linear Algebra with Applications, vol. 18, no. 5, pp. 843–855, 2011.
  • M. P. Joy and V. Tavsanoglu, “Circulant matrices and the stability of a class of CNNs,” International Journal of Circuit Theory and Applications, vol. 24, no. 1, pp. 7–13, 1996.
  • J. Delgado, N. Romero, A. Rovella, and F. Vilamajó, “Bounded solutions of quadratic circulant difference equations,” Journal of Difference Equations and Applications, vol. 11, no. 10, pp. 897–907, 2005.
  • X.-Q. Jin, V.-K. Sin, and L.-l. Song, “Circulant-block preconditioners for solving ordinary differential equations,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 409–418, 2003.
  • P. J. Davis, Circulant Matrices, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1979.
  • R. M. Gray, “Toeplitz and circulant matrices: A review,” Foundations and Trends in Communications and Information Theory, vol. 2, no. 3, pp. 155–239, 2006.
  • Z. L. Jiang and Z. X. Zhou, Circulant Matrices, Chengdu Technology University, Chengdu, China, 1999.
  • A. Bose, R. S. Hazra, and K. Saha, “Poisson convergence of eigenvalues of circulant type matrices,” Extremes, vol. 14, no. 4, pp. 365–392, 2011.
  • A. Bose, R. S. Hazra, and K. Saha, “Spectral norm of circulant-type matrices,” Journal of Theoretical Probability, vol. 24, no. 2, pp. 479–516, 2011.
  • C. Erbas and M. M. Tanik, “Generating solutions to the $N$-queens problem using $g$-circulants,” Mathematics Magazine, vol. 68, no. 5, pp. 343–356, 1995.
  • Y.-K. Wu, R.-Z. Jia, and Q. Li, “$g$-circulant solutions to the $(0,1)$ matrix equation ${A}^{m}={J}_{n}$,” Linear Algebra and Its Applications, vol. 345, pp. 195–224, 2002.
  • E. Ngondiep, S. Serra-Capizzano, and D. Sesana, “Spectral features and asymptotic properties for $g$-circulants and $g$-Toeplitz sequences,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 4, pp. 1663–1687, 2009/10.
  • S. Falcon and A. Plaza, “On $k$-Fibonacci numbers of arithmetic indexes,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 180–185, 2009.
  • Y. Yazlik and N. Taskara, “A note on generalized $k$-Horadam sequence,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 36–41, 2012.
  • S.-Q. Shen, J.-M. Cen, and Y. Hao, “On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9790–9797, 2011.
  • A. Cambini, “An explicit form of the inverse of a particular circulant matrix,” Discrete Mathematics, vol. 48, no. 2-3, pp. 323–325, 1984.
  • D. Bozkurt and T.-Y. Tam, “Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas Numbers,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 544–551, 2012.
  • W. T. Stallings and T. L. Boullion, “The pseudoinverse of an $r$-circulant matrix,” Proceedings of the American Mathematical Society, vol. 34, pp. 385–388, 1972. \endinput