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Mathematical Problems in Engineering
Volume 2014, Article ID 518913, 7 pages
http://dx.doi.org/10.1155/2014/518913
Research Article

The Explicit Identities for Spectral Norms of Circulant-Type Matrices Involving Binomial Coefficients and Harmonic Numbers

Department of Mathematics, Linyi University, Shandong 276005, China

Received 17 October 2013; Accepted 23 December 2013; Published 12 January 2014

Academic Editor: Masoud Hajarian

Copyright © 2014 Jianwei Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. C. Chu and L. D. Donno, “Hypergeometric series and harmonic number identities,” Advances in Applied Mathematics, vol. 34, no. 1, pp. 123–137, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. M. H. Ang, K. T. Arasu, S. Lun Ma, and Y. Strassler, “Study of proper circulant weighing matrices with weight 9,” Discrete Mathematics, vol. 308, no. 13, pp. 2802–2809, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. V. Brimkov, “Algorithmic and explicit determination of the Lovász number for certain circulant graphs,” Discrete Applied Mathematics, vol. 155, no. 14, pp. 1812–1825, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. A. Cambini, “An explicit form of the inverse of a particular circulant matrix,” Discrete Mathematics, vol. 48, no. 2-3, pp. 323–325, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. W. S. Chou, B. S. Du, and P. J. S. Shiue, “A note on circulant transition matrices in Markov chains,” Linear Algebra and Its Applications, vol. 429, no. 7, pp. 1699–1704, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. P. Davis, Circulant Matrices, Wiley, New York, NY, USA, 1979.
  7. C. Erbas and M. M. Tanik, “Generating solutions to the N-Queens problems using 2-circulants,” Mathematics Magazine, vol. 68, no. 5, pp. 343–356, 1995. View at Publisher · View at Google Scholar
  8. Z. L. Jiang and Z. X. Zhou, Circulant Matrices, Chengdu University of Science and Technology Press, Chengdu, China, 1999.
  9. A. Mamut, Q. X. Huang, and F. J. Liu, “Enumeration of 2-regular circulant graphs and directed double networks,” Discrete Applied Mathematics, vol. 157, no. 5, pp. 1024–1033, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. I. Stojmenović, “Multiplicative circulant networks topological properties and communication algorithms,” Discrete Applied Mathematics, vol. 77, no. 3, pp. 281–305, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. M. Ventou and C. Rigoni, “Self-dual doubly circulant codes,” Discrete Mathematics, vol. 56, no. 2-3, pp. 291–298, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. M. J. Weinberger and A. Lempel, “Factorization of symmetric circulant matrices in finite fields,” Discrete Applied Mathematics, vol. 28, no. 3, pp. 271–285, 1990. View at Google Scholar
  13. Y. K. Wu, R. Z. Jia, and Q. Li, “g-circulant solutions to the (0, 1) matrix equation Am=Jn*,” Linear Algebra and its Applications, vol. 345, no. 1–3, pp. 195–224, 2002. View at Publisher · View at Google Scholar
  14. D. Bertaccini and M. K. Ng, “Skew-circulant preconditioners for systems of LMF-based ODE codes,” in Numerical Analysis and Its Applications, Lecture Notes in Computer Science, pp. 93–101, 2001. View at Google Scholar
  15. R. Chan and X. Q. Jin, “Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems,” BIT Numerical Mathematics, vol. 31, no. 4, pp. 632–646, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. R. Chan and M. K. Ng, “Toeplitz preconditioners for Hermitian Toeplitz systems,” Linear Algebra and Its Applications C, vol. 190, pp. 181–208, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. T. Huclke, “Circulant and skew-circulant matrices for solving Toeplitz matrix problems,” SIAM Journal on Matrix Analysis and Applications, vol. 13, no. 3, pp. 767–777, 1992. View at Publisher · View at Google Scholar
  18. J. N. Lyness and T. SØrevik, “Four-dimensional lattice rules generated by skew-circulant matrices,” Mathematics of Computation, vol. 73, no. 245, pp. 279–295, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. 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. View at Publisher · View at Google Scholar · View at Scopus
  20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. A. Bose, S. Guha, R. S. Hazra, and K. Saha, “Circulant type matrices with heavy tailed entries,” Statistics and Probability Letters, vol. 81, no. 11, pp. 1706–1716, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. 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. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Solak, “On the norms of circulant matrices with the Fibonacci and Lucas numbers,” Applied Mathematics and Computation, vol. 160, no. 1, pp. 125–132, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. A. İpek, “On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 6011–6012, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  25. W. T. Stallings and T. L. Boullion, “The pseudoinverse of an r-circulant matrix,” Proceedings of the American Mathematical Society, vol. 34, no. 2, pp. 385–388, 1972. View at Google Scholar
  26. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.