Explicit form of the inverse matrices of tribonacci circulant type matrices. (English) Zbl 1383.15028

Liu, Li; Jiang, Zhaolin

Abstr. Appl. Anal. 2015, Article ID 169726, 10 p. (2015).

Summary: It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant, \(g\)-circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonacci \(g\)-circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

MSC:

15B05	Toeplitz, Cauchy, and related matrices
15B36	Matrices of integers
15A09	Theory of matrix inversion and generalized inverses

References:

[1]	Eghbali, H.; Muhaidat, S.; Hejazi, S. A.; Ding, Y., Relay selection strategies for single-carrier frequency-domain equalization multi-relay cooperative networks, IEEE Transactions on Wireless Communications, 12, 5, 2034-2045 (2013)
[2]	Sun, N.; Wu, J., Optimum sampling in spatial-temporally correlated wireless sensor networks, EURASIP Journal on Wireless Communications and Networking, 2013, article 5 (2013)
[3]	Arimoto, S., Repeat space theory applied to carbon nanotubes and related molecular networks. I, Journal of Mathematical Chemistry, 41, 3, 231-269 (2007) · Zbl 1117.92060
[4]	Chan, R. H.; Ching, W. K., Circulant preconditioners for stochastic automata networks, Numerische Mathematik, 87, 1, 35-57 (2000) · Zbl 0964.65034
[5]	Stadler, P. F.; Schnabl, W.; Forst, C. V.; Schuster, P., Dynamics of small autocatalytic reaction networks-II. Replication, mutation and catalysis, Bulletin of Mathematical Biology, 57, 1, 21-61 (1995) · Zbl 0813.92014
[6]	Jing, Y.; Jafarkhani, H., Distributed differential space-time coding for wireless relay networks, IEEE Transactions on Communications, 56, 7, 1092-1100 (2008)
[7]	Wang, G. Q.; Cheng, S. S., 6-periodic travelling waves in an artificial neural network with bang-bang control, Journal of Difference Equations and Applications, 18, 2, 261-304 (2012) · Zbl 1242.49011
[8]	Pais, D.; Caicedo-Nunez, C. H.; Leonard, N. E., Hopf bifurcations and limit cycles in evolutionary network dynamics, SIAM Journal on Applied Dynamical Systems, 11, 4, 1754-1784 (2012) · Zbl 1257.91008
[9]	Davis, P. J., Circulant Matrices (1979), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[10]	Jiang, Z. L.; Zhou, Z. X., Circulant Matrices (1999), Chengdu, China: Chengdu Technology University Publishing Company, Chengdu, China
[11]	Erbas, C.; Tanik, M. M., Generating solutions to the N-queens problem using 2-circulants, Mathematics Magazine, 68, 5, 343-356 (1995) · Zbl 0858.05002
[12]	Wu, Y. K.; Jia, R. Z.; Li, Q., \(g\)-circulant solutions to the (0, 1) matrix equation \(A^m = J_n^\star \), Linear Algebra and Its Applications, 345, 1-3, 195-224 (2002) · Zbl 0998.15015
[13]	Ngondiep, E.; Serra-Capizzano, S.; Sesana, D., Spectral features and asymptotic properties for \(g\)-circulants and \(g\)-Toeplitz sequences, SIAM Journal on Matrix Analysis and Applications, 31, 4, 1663-1687 (2010) · Zbl 1206.15010
[14]	Jiang, Z. L.; Gong, Y. P.; Gao, Y., Circulant type matrices with the sum and product of Fibonacci and Lucas numbers, Abstract and Applied Analysis, 2014 (2014) · Zbl 1472.15007
[15]	Jiang, Z.; Gong, Y.; Gao, Y., Invertibility and explicit inverses of circulant-type matrices with \(k\)-Fibonacci and \(k\)-Lucas numbers, Abstract and Applied Analysis, 2014 (2014) · Zbl 1473.15033
[16]	Jiang, X.; Hong, K., Exact determinants of some special circulant matrices involving four kinds of famous numbers, Abstract and Applied Analysis, 2014 (2014) · Zbl 1472.15006
[17]	Bozkurt, D.; Tam, T. Y., Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas Numbers, Applied Mathematics and Computation, 219, 2, 544-551 (2012) · Zbl 1302.15005
[18]	Cambini, A., An explicit form of the inverse of a particular circulant matrix, Discrete Mathematics, 48, 2-3, 323-325 (1984) · Zbl 0531.15001
[19]	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
[20]	Elia, M., Derived sequences, the Tribonacci recurrence and cubic forms, The Fibonacci Quarterly, 39, 2, 107-115 (2001) · Zbl 1001.11008
[21]	Balof, B., Restricted tilings and bijections, Journal of Integer Sequences, 15 (2012) · Zbl 1292.05017
[22]	Rabinowitz, S., Algorithmic manipulation of third-order linear recurrences, The Fibonacci Quarterly, 34, 5, 447-464 (1996) · Zbl 0872.11009
[23]	Li, J.; Jiang, Z. L.; Lu, F., Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers, Abstract and Applied Analysis, 2014 (2014) · Zbl 1473.15013
[24]	Ding, D.; Wang, Z.; Hu, J.; Shu, H., Dissipative control for state-saturated discrete time-varying systems with randomly occurring nonlinearities and missing measurements, International Journal of Control, 86, 4, 674-688 (2013) · Zbl 1278.93279
[25]	Wang, Z.; Dong, H.; Shen, B.; Gao, H., Finite-horizon \(H_\infty\) filtering with missing measurements and quantization effects, IEEE Transactions on Automatic Control, 58, 7, 1707-1718 (2013) · Zbl 1369.93660
[26]	Hu, J.; Wang, Z.; Shen, B.; Gao, H., Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements, International Journal of Control, 86, 4, 650-663 (2013) · Zbl 1278.93269