## Journal of Applied Mathematics

### Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers

#### Abstract

The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 585438, 11 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.jam/1395855294

Digital Object Identifier
doi:10.1155/2014/585438

Mathematical Reviews number (MathSciNet)
MR3166775

Zentralblatt MATH identifier
07010687

#### Citation

Jiang, Zhaolin; Shen, Nuo; Li, Juan. Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers. J. Appl. Math. 2014 (2014), Article ID 585438, 11 pages. doi:10.1155/2014/585438. https://projecteuclid.org/euclid.jam/1395855294

#### References

• F. Zanderigo, A. Bertoldo, G. Pillonetto, and C. Cobelli, “Nonlinear stochastic regularization to characterize tissue residue function in bolus-tracking MRI: assessment and comparison with SVD, block-circulant SVD, and Tikhonov,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 5, pp. 1287–1297, 2009.
• 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.
• W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” in Visual Communications and Image Processing, vol. 7744 of Proceedings of SPIE, July 2010.
• O. Wu, L. ${\text{\O}}$stergaard, R. M. Weisskoff, T. Benner, B. R. Rosen, and A. G. Sorensen, “Tracer arrival timing-insensitive technique for estimating flow in MR perfusion-weighted imaging using singular value decomposition with a block-circulant deconvolution matrix,” Magnetic Resonance in Medicine, vol. 50, no. 1, pp. 164–174, 2003.
• 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. Chillag, “Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes,” Linear Algebra and Its Applications, vol. 218, pp. 147–183, 1995.
• Z.-L. Jiang and Z.-B. Xu, “Efficient algorithm for finding the inverse and the group inverse of FLS $r$-circulant matrix,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 45–57, 2005.
• J. Li, Z. Jiang, and N. Shen, “Explicit determinants of the Fibonacci RFPLR circulant and Lucas RFPLR circulant matrix,” JP Journal of Algebra, Number Theory and Applications, vol. 28, no. 2, pp. 167–179, 2013.
• Z. L. Jiang, J. Li, and N. Shen, “On the explicit determinants of the RFPLR and RFPLL circulant matrices involving Pell numbers in information theory,” in Proceedings of the International Conference on Information and Computer Applications (ICICA '12), vol. 308 of Communications in Computer and Information Science, pp. 364–370, 2012.
• Z. Tian, “Fast algorithms for solving the inverse problem of $AX=b$ in four different families of patterned matrices,” Far East Journal of Applied Mathematics, vol. 52, no. 1, pp. 1–12, 2011.
• N. Shen, Z. L. Jiang, and J. Li, “On explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers,” WSEAS Transactions on Mathematics, vol. 12, pp. 42–53, 2013.
• D. V. Jaiswal, “On determinants involving generalized Fibonacci numbers,” The Fibonacci Quarterly, vol. 7, pp. 319–330, 1969.
• D. Lin, “Fibonacci-Lucas quasi-cyclic matrices,” The Fibonacci Quarterly, vol. 40, no. 3, pp. 280–286, 2002.
• D. A. Lind, “A Fibonacci circulant,” The Fibonacci Quarterly, vol. 8, no. 5, pp. 449–455, 1970.
• 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.
• 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.
• F. Yilmaz and D. Bozkurt, “Hessenberg matrices and the Pell and Perrin numbers,” Journal of Number Theory, vol. 131, no. 8, pp. 1390–1396, 2011.
• J. Blümlein, D. J. Broadhurst, and J. A. M. Vermaseren, “The multiple zeta value data mine,” Computer Physics Communications, vol. 181, no. 3, pp. 582–625, 2010.
• M. Janjić, “Determinants and recurrence sequences,” Journal of Integer Sequences, vol. 15, no. 3, pp. 1–12, 2012.
• M. Elia, “Derived sequences, the Tribonacci recurrence and cubic forms,” The Fibonacci Quarterly, vol. 39, no. 2, pp. 107–115, 2001. \endinput