## Abstract and Applied Analysis

### The Invertibility, Explicit Determinants, and Inverses of Circulant and Left Circulant and $g$-Circulant Matrices Involving Any Continuous Fibonacci and Lucas Numbers

#### Abstract

Circulant matrices play an important role in solving delay differential equations. In this paper, circulant type matrices including the circulant and left circulant and $g$-circulant matrices with any continuous Fibonacci and Lucas numbers are considered. Firstly, the invertibility of the circulant matrix is discussed and the explicit determinant and the inverse matrices by constructing the transformation matrices are presented. Furthermore, the invertibility of the left circulant and $g$-circulant matrices is also studied. We obtain the explicit determinants and the inverse matrices of the left circulant and $g$-circulant matrices by utilizing the relationship between left circulant, $g$-circulant matrices and circulant matrix, respectively.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 931451, 14 pages.

Dates
First available in Project Euclid: 3 October 2014

https://projecteuclid.org/euclid.aaa/1412360626

Digital Object Identifier
doi:10.1155/2014/931451

Mathematical Reviews number (MathSciNet)
MR3246367

Zentralblatt MATH identifier
07023336

#### Citation

Jiang, Zhaolin; Li, Dan. The Invertibility, Explicit Determinants, and Inverses of Circulant and Left Circulant and $g$ -Circulant Matrices Involving Any Continuous Fibonacci and Lucas Numbers. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 931451, 14 pages. doi:10.1155/2014/931451. https://projecteuclid.org/euclid.aaa/1412360626

#### References

• D. Bertaccini, “A circulant preconditioner for the systems of LMF-based ODE codes,” SIAM Journal on Scientific Computing, vol. 22, no. 3, pp. 767–786, 2000.
• Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, Mass, USA, 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.
• R. H. Chan and M. K. Ng, “Conjugate gradient methods for Toeplitz systems,” SIAM Review, vol. 38, no. 3, pp. 427–482, 1996.
• X. Q. Jin, Developments and Applications of Block Toeplitz Iterative Solvers, Kluwer Academic, Dodrecht, The Netherlands, 2002.
• R. H. Chan, M. K. Ng, and X. Q. Jin, “Strang-type preconditioners for systems of LMF-based ODE codes,” IMA Journal of Numerical Analysis, vol. 21, no. 2, pp. 451–462, 2001.
• R. D. Nussbaum, “Circulant matrices and differential-delay equations,” Journal of Differential Equations, vol. 60, no. 2, pp. 201–217, 1985.
• Z. J. Bai, X. Q. Jin, and L. L. Song, “Strang-type preconditioners for solving linear systems from neutral delay differential equations,” Calcolo, vol. 40, no. 1, pp. 21–31, 2003.
• R. H. Chan, X. Jin, and Y. Tam, “Strang-type preconditioners for solving systems of ODEs by boundary value methods,” Electronic Journal of Mathematical and Physical Sciences, vol. 1, no. 1, pp. 14–46, 2002.
• X. Q. Jin, S. L. Lei, and Y. Wei, “Circulant preconditioners for solving differential equations with multidelays,” Computers & Mathematics with Applications, vol. 47, no. 8-9, pp. 1429–1436, 2004.
• S. L. Lei and X. Q. Jin, “Strang-type preconditioners for solving differential-algebraic equations,” in Numerical Analysis and Its Applications, L. Vulkov, J. Wasniewski, and P. Yalamov, Eds., Lecture Notes in Computer Science, pp. 505–512, Springer, Berlin, Germany, 2001.
• F. R. Lin, X. Q. Jin, and S. L. Lei, “Strang-type preconditioners for solving linear systems from delay differential equations,” BIT Numerical Mathematics, vol. 43, no. 1, pp. 139–152, 2003.
• S. J. Guo, Y. M. Chen, and J. H. Wu, “Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory,” Acta Mathematica Sinica, vol. 28, no. 4, pp. 825–856, 2012.
• M. Cai and X. Jin, “BCCB preconditioners for solving linear systems from delay differential equations,” Computers and Mathematics with Applications, vol. 50, no. 1-2, pp. 281–288, 2005.
• X. Jin, S. Lei, and Y. Wei, “Circulant preconditioners for solving singular perturbation delay differential equations,” Numerical Linear Algebra with Applications, vol. 12, no. 2-3, pp. 327–336, 2005.
• C. Erbas and M. M. Tanik, “Generating solutions to the \emphN-queens problem using 2-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 \emphA$^{m}$ = \emphJ$^{\ast\,\!}_{n}$,” Linear Algebra and Its Applications, vol. 345, pp. 195–224, 2002.
• A. Bose, R. S. Hazra, and K. Saha, “Poisson convergence of eigenvalues of circulant type matrices,” Extremes: Statistical Theory and Applications in Science, Engineering and Economics, 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.
• 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, 2010.
• P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, NY, USA, 1979.
• Z. L. Jiang and Z. X. Zhou, Circulant Matrices, Chengdu Technology University Publishing Company, Chengdu, China, 1999.
• D. V. Jaiswal, “On determinants involving generalized Fibonacci numbers,” The Fibonacci Quarterly, vol. 7, pp. 319–330, 1969.
• D. A. Lind, “A Fibonacci circulant,” The Fibonacci Quarterly, vol. 8, no. 5, pp. 449–455, 1970.
• D. Z. Lin, “Fibonacci-Lucas quasi-cyclic matrices,” The Fibonacci Quarterly, vol. 40, no. 3, pp. 280–286, 2002.
• S. Shen, J. 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.
• D. Bozkurt and T. Tam, “Determinants and inverses of circula nt matrices with Jacobsthal and Jacobsthal-Lucas numbers,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 544–551, 2012.
• H. Karner, J. Schneid, and C. Ueberhuber, “Spectral decomposition of real circulant matrices,” Linear Algebra and Its Applications, vol. 367, pp. 301–311, 2003.
• 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