Exact inverse matrices of Fermat and Mersenne circulant matrix. (English) Zbl 1383.15029

Summary: The well known circulant matrices are applied to solve networked systems. In this paper, circulant and left circulant matrices with the Fermat and Mersenne numbers are considered. The nonsingularity of these special matrices is discussed. Meanwhile, the exact determinants and inverse matrices of these special matrices are presented.


15B05 Toeplitz, Cauchy, and related matrices
15A09 Theory of matrix inversion and generalized inverses
Full Text: DOI


[1] Zhang, C.; Dangelmayr, G.; Oprea, I., Storing cycles in Hopfield-type networks with pseudoinverse learning rule: admissibility and network topology, Neural Networks, 46, 283-298 (2013) · Zbl 1296.68154
[2] Rocchesso, D.; Smith, J. O., Circulant and elliptic feedback delay networks for artificial reverberation, IEEE Transactions on Speech and Audio Processing, 5, 1, 51-63 (1997)
[3] Jing, Y.; Jafarkhani, H., Distributed differential space-time coding for wireless relay networks, IEEE Transactions on Communications, 56, 7, 1092-1100 (2008)
[4] Bašić, M., Characterization of quantum circulant networks having perfect state transfer, Quantum Information Processing, 12, 1, 345-364 (2013) · Zbl 1309.81043
[5] Wang, G.; Gao, F.; Wu, Y.-C.; Tellambura, C., Joint CFO and channel estimation for OFDM-based two-way relay networks, IEEE Transactions on Wireless Communications, 10, 2, 456-465 (2011)
[6] Li, J.; Yuan, J.; Malaney, R.; Xiao, M.; Chen, W., Full-diversity binary frame-wise network coding for multiple-source multiple-relay networks over slow-fading channels, IEEE Transactions on Vehicular Technology, 61, 3, 1346-1360 (2012)
[7] Davis, P. J., Circulant Matrices (1979), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[8] Jiang, Z. L.; Zhou, Z. X., Circulant Matrices (1999), Chengdu, China: Chengdu Technology University Publishing Company, Chengdu, China
[9] Jiang, Z., On the minimal polynomials and the inverses of multilevel scaled factor circulant matrices, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.15072
[10] Jiang, Z.; Xu, T.; Lu, F., Isomorphic operators and functional equations for the skew-circulant algebra, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.15016
[11] Li, J.; Jiang, Z. L.; Lu, F. L., Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers, Abstract and Applied Analysis, 2014 (2014) · Zbl 1473.15013
[12] Jiang, Z. L., Nonsingularity of two classes of cyclic matrices, Mathematics in Practice and Theory, 2, 52-58 (1995) · Zbl 1493.15121
[13] Yao, J.-J.; Jiang, Z.-L., The determinants, inverses, norm, and spread of skew circulant type matrices involving any continuous Lucas numbers, Journal of Applied Mathematics, 2014 (2014) · Zbl 1437.15052
[14] 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
[15] Dazheng, L., Fibonacci-Lucas quasi-cyclic matrices, The Fibonacci Quarterly, 40, 3, 280-286 (2002) · Zbl 1081.11011
[16] 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
[17] Horadam, A. F., Further appearence of the Fibonacci sequence, The Fibonacci Quarterly, 1, 4, 41-42 (1963)
[18] İpek, A.; Arı, K., On Hessenberg and pentadiagonal determinants related with FIBonacci and FIBonacci-like numbers, Applied Mathematics and Computation, 229, 433-439 (2014) · Zbl 1364.11041
[19] Dong, H.; Wang, Z.; Gao, H., Distributed \(H_∞\) filtering for a class of markovian jump nonlinear time-delay systems over lossy sensor networks, IEEE Transactions on Industrial Electronics, 60, 10, 4665-4672 (2013)
[20] 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
[21] 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
[22] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.