Journal of Applied Mathematics

A Note on the Square Roots of a Class of Circulant Matrices

Ying Zhang, Huisheng Zhang, and Guoyan Chen

Full-text: Open access

Abstract

We prove that any k-circulant matrix and any even order skew k-circulant matrix are diagonalizable for any k. Then, we propose two algorithms for computing the square roots of the k-circulant matrix and the skew k-circulant matrix, respectively. In particular, we show that the square roots of the k-circulant matrix are still k-circulant matrices. Both the theoretical analysis and the numerical experiments show that our algorithms are faster than the standard Schur method.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 601243, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808222

Digital Object Identifier
doi:10.1155/2013/601243

Mathematical Reviews number (MathSciNet)
MR3130993

Zentralblatt MATH identifier
06950775

Citation

Zhang, Ying; Zhang, Huisheng; Chen, Guoyan. A Note on the Square Roots of a Class of Circulant Matrices. J. Appl. Math. 2013 (2013), Article ID 601243, 6 pages. doi:10.1155/2013/601243. https://projecteuclid.org/euclid.jam/1394808222


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References

  • N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, Pa, USA, 2008.
  • N. J. Higham, “Newton's method for the matrix square root,” Mathematics of Computation, vol. 46, no. 174, pp. 537–549, 1986.
  • N. J. Higham, “Stable iterations for the matrix square root,” Numerical Algorithms, vol. 15, no. 2, pp. 227–242, 1997.
  • E. D. Denman, “Roots of real matrices,” Linear Algebra and Its Applications, vol. 36, pp. 133–139, 1981.
  • L. S. Shieh and N. Chahin, “A computer-aided method for the factorization of matrix polynomials,” International Journal of Systems Science, vol. 12, no. 3, pp. 305–323, 1981.
  • E. D. Denman and A. N. Beavers, Jr., “The matrix sign function and computations in systems,” Applied Mathematics and Computation, vol. 2, no. 1, pp. 63–94, 1976.
  • A. Björck and S. Hammarling, “A Schur method for the square root of a matrix,” Linear Algebra and Its Applications, vol. 52-53, pp. 127–140, 1983.
  • W. D. Hoskins and D. J. Walton, “A fast method of computing the square root of a matrix,” IEEE Transactions on Automatic Control, vol. 23, no. 3, pp. 494–495, 1978.
  • W. D. Hoskins and D. J. Walton, “A faster, more stable method for computing the $p$th roots of positive definite matrices,” Linear Algebra and Its Applications, vol. 26, pp. 139–163, 1979.
  • W. Zhao, “The inverse problem of anti-circulant matrices in signal processing,” in Proceedings of the Pacific-Asia Conference on Knowledge Engineering and Software Engineering, pp. 47–50, Shenzhen, China, 2009.
  • G. Zhao, “The improved nonsingularity on the r-circulant mat-rices in signal processing,” in Proceedings of the International Conference on Computer Techology and Development, pp. 564–567, Kota Kinabalu, Malaysia, 2009.
  • V. C. Liu and P. P. Vaidyanathan, “Circulant and skew-circulant matrices as new normal-form realization of IIR digital filters,” IEEE Transactions on Circuits and Systems, vol. 35, no. 6, pp. 625–635, 1988.
  • A. Mayer, A. Castiaux, and J.-P. Vigneron, “Electronic Green scattering with $n$-fold symmetry axis from block circulant mat-rices,” Computer Physics Communications, vol. 109, no. 1, pp. 81–89, 1998.
  • M. K. Ng, “Circulant and skew-circulant splitting methods for Toeplitz systems,” Journal of Computational and Applied Math-ematics, vol. 159, no. 1, pp. 101–108, 2003.
  • C. Lu, “On the logarithms of circulant matrices,” Journal of Computational Analysis and Applications, vol. 15, no. 3, pp. 402–412, 2013.
  • C. Lu and C. Gu, “The computation of the square roots of circulant matrices,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6819–6829, 2011.
  • L. Lin and Z.-Y. Liu, “On the square root of an $H$-matrix with positive diagonal elements,” Annals of Operations Research, vol. 103, pp. 339–350, 2001.
  • Y. Mei, “Computing the square roots of a class of circulant matrices,” Journal of Applied Mathematics, Article ID 647623, 15 pages, 2012.
  • R. H. Chan and M. K. Ng, “Conjugate gradient methods for Toeplitz systems,” SIAM Review, vol. 38, no. 3, pp. 427–482, 1996.
  • P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, NY, USA, 1979.
  • J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Mathematics of Computation, vol. 19, pp. 297–301, 1965.
  • C. Lu and C. Gu, “The computation of the inverse of block-wise centrosymmetric matrices,” Publicationes Mathematicae Debrecen, vol. 82, no. 2, pp. 379–397, 2013.
  • T. Finck, G. Heinig, and K. Rost, “An inversion formula and fast algorithms for Cauchy-Vandermonde matrices,” Linear Algebra and Its Applications, vol. 183, pp. 179–191, 1993.
  • I. Gohberg and V. Olshevsky, “Complexity of multiplication with vectors for structured matrices,” Linear Algebra and Its Applications, vol. 202, pp. 163–192, 1994.
  • J. R. Bunch and J. E. Hopcroft, “Triangular factorization and inversion by fast matrix multiplication,” Mathematics of Computation, vol. 28, pp. 231–236, 1974.
  • C. H. Papadimitriou, Computational Complexity, John Wiley & Sons, Chichester, UK, 2003.