Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers. (English) Zbl 1473.15013

Li, Juan; Jiang, Zhaolin; Lu, Fuliang

Abstr. Appl. Anal. 2014, Article ID 381829, 9 p. (2014).

Summary: Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degree \(n\), the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

MSC:

15A15	Determinants, permanents, traces, other special matrix functions
11C20	Matrices, determinants in number theory
11B39	Fibonacci and Lucas numbers and polynomials and generalizations

Keywords:

tribonacci numbers; Lucas numbers; determinant

References:

[1]	Chen, W.; Lin, J.; Chen, C. S., The method of fundamental solutions for solving exterior axisymmetric Helmholtz problems with high wave-number, Advances in Applied Mathematics and Mechanics, 5, 4, 477-493 (2013) · Zbl 1279.35013
[2]	Lei, S.-L.; Sun, H.-W., A circulant preconditioner for fractional diffusion equations, Journal of Computational Physics, 242, 715-725 (2013) · Zbl 1297.65095 · doi:10.1016/j.jcp.2013.02.025
[3]	Karasozen, B.; Simsek, G., Energy preserving integration of bi-Hamiltonian partial differential equations, Applied Mathematics Letters, 26, 12, 1125-1133 (2013) · Zbl 1308.35249
[4]	Otto, K., Analysis of preconditioners for hyperbolic partial differential equations, SIAM Journal on Numerical Analysis, 33, 6, 2131-2165 (1996) · Zbl 0861.65072
[5]	Sachs, E. W.; Strauss, A. K., Efficient solution of a partial integro-differential equation in finance, Applied Numerical Mathematics, 58, 11, 1687-1703 (2008) · Zbl 1155.65109 · doi:10.1016/j.apnum.2007.11.002
[6]	Gilmour, A. E., Circulant matrix methods for the numerical solution of partial differential equations by FFT convolutions, Applied Mathematical Modelling, 12, 1, 44-50 (1988) · Zbl 0645.65065
[7]	Brockett, R. W.; Willems, J. L., Discretized partial differential equations: examples of control systems defined on modules, Automatica, 10, 5, 507-515 (1974) · Zbl 0288.93017
[8]	Cohn, S. E.; Dee, D. P., Observability of discretized partial differential equations, SIAM Journal on Numerical Analysis, 25, 3, 586-617 (1988) · Zbl 0657.93009 · doi:10.1137/0725037
[9]	Zhang, C.; Chen, H.; Wang, L., Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations, Numerical Linear Algebra with Applications, 18, 5, 843-855 (2011) · Zbl 1249.65165 · doi:10.1002/nla.770
[10]	Delgado, J.; Romero, N.; Rovella, A.; Vilamajó, F., Bounded solutions of quadratic circulant difference equations, Journal of Difference Equations and Applications, 11, 10, 897-907 (2005) · Zbl 1097.37011 · doi:10.1080/10236190500138312
[11]	Jin, X.-Q.; Sin, V.-K.; Song, L.-L., Circulant-block preconditioners for solving ordinary differential equations, Applied Mathematics and Computation, 140, 2-3, 409-418 (2003) · Zbl 1044.65062 · doi:10.1016/S0096-3003(02)00237-0
[12]	Wilde, A. C., Differential equations involving circulant matrices, The Rocky Mountain Journal of Mathematics, 13, 1, 1-13 (1983) · Zbl 0516.34016 · doi:10.1216/RMJ-1983-13-1-1
[13]	Wittsack, H.-J.; Wohlschläger, A. M.; Ritzl, E. K.; Kleiser, R.; Cohnen, M.; Seitz, R. J.; Mödder, U., CT-perfusion imaging of the human brain: advanced deconvolution analysis using circulant singular value decomposition, Computerized Medical Imaging and Graphics, 32, 1, 67-77 (2008) · doi:10.1016/j.compmedimag.2007.09.004
[14]	Henriques, J. F.; Caseiro, R.; Martins, P.; Batista, J., Exploiting the circulant structure of tracking-by-detection with kernels, Proceedings of the European Conference on Computer Vision (ECCV ’12)
[15]	Huang, X.; Ye, G.; Wong, K.-W., Chaotic image encryption algorithm based on circulant operation, Abstract and Applied Analysis, 2013 (2013) · Zbl 1381.94019 · doi:10.1155/2013/384067
[16]	Andrecut, M., Applications of left circulant matrices in signal and image processing, Modern Physics Letters B, 22, 4, 231-241 (2008) · Zbl 1158.94310 · doi:10.1142/S0217984908014791
[17]	Davis, P. J., Circulant Matrices (1979), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0418.15017
[18]	Jiang, Z. L.; Zhou, Z. X., Circulant Matrices (1999), Chengdu, China: Chengdu Technology University, Chengdu, China
[19]	Jaiswal, D. V., On determinants involving generalized Fibonacci numbers, The Fibonacci Quarterly, 7, 319-330 (1969) · Zbl 0191.04501
[20]	Lind, D. A., A Fibonacci circulant, The Fibonacci Quarterly, 8, 5, 449-455 (1970) · Zbl 0215.06802
[21]	Lin, D., Fibonacci-Lucas quasi-cyclic matrices, The Fibonacci Quarterly, 40, 3, 280-286 (2002) · Zbl 1081.11011
[22]	Shen, S.-Q.; Cen, J.-M.; Hao, Y., On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Applied Mathematics and Computation, 217, 23, 9790-9797 (2011) · Zbl 1222.15010 · doi:10.1016/j.amc.2011.04.072
[23]	Gao, Y.; Jiang, Z. L.; Gong, Y. P., On the determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas Numbers, WSEAS Transactions on Mathematics, 12, 4, 472-481 (2013)
[24]	Jiang, X. Y.; Gao, Y.; Jiang, Z. L., Determinants and inverses of skew and skew left circulant matrices involving the k-Fibonacci numbers in communications-I, Far East Journal of Mathematical Sciences, 76, 1, 123-137 (2013) · Zbl 1276.15005
[25]	Jiang, X. Y.; Gao, Y.; Jiang, Z. L., Determinants and inverses of skew and skew left circulant matrices involving the k-Lucas numbers in communications-II, Far East Journal of Mathematical Sciences, 78, 1, 1-17 (2013) · Zbl 1390.15021
[26]	Solak, S., On the norms of circulant matrices with the Fibonacci and Lucas numbers, Applied Mathematics and Computation, 160, 1, 125-132 (2005) · Zbl 1066.15029 · doi:10.1016/j.amc.2003.08.126
[27]	Ipek, A., On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Applied Mathematics and Computation, 217, 12, 6011-6012 (2011) · Zbl 1211.15028 · doi:10.1016/j.amc.2010.12.094
[28]	Shen, S.; Cen, J., On the bounds for the norms of \(r\)-circulant matrices with the Fibonacci and Lucas numbers, Applied Mathematics and Computation, 216, 10, 2891-2897 (2010) · Zbl 1211.15029 · doi:10.1016/j.amc.2010.03.140
[29]	Akbulak, M.; Bozkurt, D., On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacettepe Journal of Mathematics and Statistics, 37, 2, 89-95 (2008) · Zbl 1204.15031
[30]	Bose, A.; Hazra, R. S.; Saha, K., Spectral norm of circulant-type matrices, Journal of Theoretical Probability, 24, 2, 479-516 (2011) · Zbl 1241.60006 · doi:10.1007/s10959-009-0257-z
[31]	Mirsky, L., The spread of a matrix, Mathematika, 3, 127-130 (1956) · Zbl 0073.00903
[32]	Sharma, R.; Kumar, R., Remark on upper bounds for the spread of a matrix, Linear Algebra and its Applications, 438, 11, 4359-4362 (2013) · Zbl 1281.15018 · doi:10.1016/j.laa.2013.01.018
[33]	Wu, J.; Zhang, P.; Liao, W., Upper bounds for the spread of a matrix, Linear Algebra and its Applications, 437, 11, 2813-2822 (2012) · Zbl 1258.15004 · doi:10.1016/j.laa.2012.07.007
[34]	Johnson, C. R.; Kumar, R.; Wolkowicz, H., Lower bounds for the spread of a matrix, Linear Algebra and Its Applications, 71, 161-173 (1985) · Zbl 0578.15013
[35]	Elia, M., Derived sequences, the Tribonacci recurrence and cubic forms, The Fibonacci Quarterly, 39, 2, 107-115 (2001) · Zbl 1001.11008
[36]	Balof, B., Restricted tilings and bijections, Journal of Integer Sequences, 15 (2012) · Zbl 1292.05017
[37]	Rabinowitz, S., Algorithmic manipulation of third-order linear recurrences, The Fibonacci Quarterly, 34, 5, 447-464 (1996) · Zbl 0872.11009
[38]	Bose, A.; Mitra, J., Limiting spectral distribution of a special circulant, Statistics and Probability Letters, 60, 1, 111-120 (2002) · Zbl 1014.60038 · doi:10.1016/S0167-7152(02)00289-4