## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2014, Special Issue (2013), Article ID 239693, 10 pages.

### The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

#### Abstract

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

#### Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2013), Article ID 239693, 10 pages.

Dates
First available in Project Euclid: 1 October 2014

https://projecteuclid.org/euclid.jam/1412177185

Digital Object Identifier
doi:10.1155/2014/239693

Mathematical Reviews number (MathSciNet)
MR3198369

#### Citation

Yao, Jin-jiang; Jiang, Zhao-lin. The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers. J. Appl. Math. 2014, Special Issue (2013), Article ID 239693, 10 pages. doi:10.1155/2014/239693. https://projecteuclid.org/euclid.jam/1412177185

#### References

• A. Daher, E. H. Baghious, and G. Burel, “Fast algorithm for optimal design of block digital filters based on circulant matrices,” IEEE Signal Processing Letters, vol. 15, pp. 637–640, 2008.
• 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.
• D. Q. Fu, Z. L. Jiang, Y. F. Cui, and S. T. Jhang, “A new fastalgorithm for optimal design of block digital filters by skew-cyclic convolution,” IET Signal Processing, p. 6, 2014.
• H.-J. Wittsack, A. M. Wohlschläger, E. K. Ritzl et al., “CT-perfusion imaging of the human brain: advanced deconvolution analysis using circulant singular value decomposition,” Computerized Medical Imaging and Graphics, vol. 32, no. 1, pp. 67–77, 2008.
• J. F. Henriques, R. Caseiro, P. Martins, and J. Batista, “Exploiting the circulant structure of tracking-by-detection with kernels,” in Computer Vision–-ECCV 2012, vol. 7575 of Lecture Notes in Computer Science, pp. 702–715, Springer, Berlin, Germany, 2012.
• X. Huang, G. Ye, and K.-W. Wong, “Chaotic image encryption algorithm based on circulant operation,” Abstract and Applied Analysis, vol. 2013, Article ID 384067, 8 pages, 2013.
• Y. Jing and H. Jafarkhani, “Distributed differential space-time coding for wireless relay networks,” IEEE Transactions on Communications, vol. 56, no. 7, pp. 1092–1100, 2008.
• M. J. Narasimha, “Linear convolution using skew-cyclic convolutions,” IEEE Signal Processing Letters, vol. 14, no. 3, pp. 173–176, 2007.
• T. A. Gulliver and M. Harada, “New nonbinary self-dual codes,” IEEE Transactions on Information Theory, vol. 54, no. 1, pp. 415–417, 2008.
• 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, Chengdu, China, 1999.
• D. Bertaccini and M. K. Ng, “Skew-circulant preconditioners for systems of LMF-based ODE codes,” in Numerical Analysis and Its Applications, vol. 1988 of Lecture Notes in Computer Science, pp. 93–101, Springer, Berlin, Germany, 2001.
• R. H. 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.
• R. H. Chan and K.-P. Ng, “Toeplitz preconditioners for Hermitian Toeplitz systems,” Linear Algebra and Its Applications, vol. 190, pp. 181–208, 1993.
• T. Huckle, “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.
• 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.
• H. Karner, J. Schneid, and C. W. Ueberhuber, “Spectral decomposition of real circulant matrices,” Linear Algebra and Its Applications, vol. 367, pp. 301–311, 2003.
• J. Li, Z. L. Jiang, N. Shen, and J. W. Zhou, “On optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix,” Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 707381, 7 pages, 2013.
• 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.
• L. Dazheng, “Fibonacci-Lucas quasi-cyclic matrices,” The Fibonacci Quarterly, vol. 40, no. 3, pp. 280–286, 2002.
• 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.
• Y. Gao, Z. L. Jiang, and Y. P. Gong, “On the determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas numbers,” WSEAS Transactions on Mathematics, vol. 12, no. 4, pp. 472–481, 2013.
• X. Y. Jiang, Y. Gao, and Z. L. Jiang, “Determinantsand inverses of skew and skew left circulant matrices involving the $k$-Fibonacci numbers in communications-I,” Far East Journal of Mathematical Sciences, vol. 76, no. 1, pp. 123–137, 2013.
• X. Y. Jiang, Y. Gao, and Z. L. Jiang, “Determinants and inverses of skew and skew left circulant matrices involving the $k$-Lucas numbers in communications-II,” Far East Journal of Mathematical Sciences, vol. 78, no. 1, pp. 1–17, 2013.
• 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.
• A. \.Ipek, “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.
• S. Shen and J. Cen, “On the bounds for the norms of $r$-circulant matrices with the Fibonacci and Lucas numbers,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2891–2897, 2010.
• M. Akbulak and D. Bozkurt, “On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers,” Hacettepe Journal of Mathematics and Statistics, vol. 37, no. 2, pp. 89–95, 2008.
• 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.
• L. Mirsky, “The spread of a matrix,” Mathematika, vol. 3, pp. 127–130, 1956.
• R. Sharma and R. Kumar, “Remark on upper bounds for the spread of a matrix,” Linear Algebra and Its Applications, vol. 438, no. 11, pp. 4359–4362, 2013.
• L. Mirsky, “Inequalities for normal and Hermitian matrices,” Duke Mathematical Journal, vol. 24, no. 4, pp. 591–599, 1957.
• E. R. Barnes and A. J. Hoffman, “Bounds for the spectrum of normal matrices,” Linear Algebra and Its Applications, vol. 201, pp. 79–90, 1994.
• E. Jiang and X. Zhan, “Lower bounds for the spread of a Hermitian matrix,” Linear Algebra and Its Applications, vol. 256, pp. 153–163, 1997.
• R. Bhatia and R. Sharma, “Some inequalities for positive linear maps,” Linear Algebra and Its Applications, vol. 436, no. 6, pp. 1562–1571, 2012.
• J. Wu, P. Zhang, and W. Liao, “Upper bounds for the spread of a matrix,” Linear Algebra and Its Applications, vol. 437, no. 11, pp. 2813–2822, 2012.
• C. R. Johnson, R. Kumar, and H. Wolkowicz, “Lower bounds for the spread of a matrix,” Linear Algebra and Its Applications, vol. 71, pp. 161–173, 1985. \endinput