## Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers.(English)Zbl 1302.15005

Summary: Let $$n\geq 3$$ and let $$\mathbb J_n=\mathrm{circ}(J_1,J_2,\dots,J_n)$$ and $$\mathbb j_n = \mathrm{circ}(j_1,j_2,\ldots,j_{n-1})$$, be the $$n\times n$$ circulant matrices associated with the Jacobsthal numbers $$J_1,\dots,J_n$$ and the Jacobsthal-Lucas numbers $$j_1,\ldots,j_{n-1}$$, respectively. The determinants and the inverses of $$J_{n}$$ and $$j_{n}$$ are obtained in terms of $$J_1,\dots,J_n$$ and $$j_1,\ldots,j_{n-1}$$, respectively.

### MSC:

 15A09 Theory of matrix inversion and generalized inverses 15B36 Matrices of integers 15A15 Determinants, permanents, traces, other special matrix functions
Full Text:

### References:

  Akbulak, M.; Bozkurt, D., On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacettepe J. math. stat., 37, 2, 89-95, (2008) · Zbl 1204.15031  Aldrovandi, R., Special matrices of mathematical physics: stochastic circulant and Bell matrices, (2001), World Scientific Singapore · Zbl 0982.15040  Davis, P.J., Circulant matrices, (1979), Wiley NewYork · Zbl 0418.15017  Frey, D.D.; Sellers, J.A., Jacobsthal numbers and alternating sign matrices, J. integer sequences, 3, 1-15, (2000) · Zbl 0961.15008  Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge Univ. Press Cambridge · Zbl 0576.15001  Lee, G.Y.; Kim, J.S.; Lee, S.G., Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci quart., 40, 203-211, (2002) · Zbl 1079.11012  Miladinovic, M.; Stanimirovic, P., Singular case of generalized Fibonacci and Lucas numbers, J. koeran math. soc., 48, 33-48, (2011) · Zbl 1226.05017  Shen, S.Q.; Cen, J.M., On the bounds for the norms of r-circulant matrices with Fibonacci and Lucas numbers, Appl. math. comput., 216, 2891-2897, (2010) · Zbl 1211.15029  Shen, S.Q.; Cen, J.M.; Hao, Y., On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Appl. math. comput., 217, 9790-9797, (2011) · Zbl 1222.15010  Solak, S., On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. math. comput., 160, 125-132, (2005) · Zbl 1066.15029  Stanimirović, P.; Miladinović, M., Inversion of the generalized Fibonacci matrix by convolution, Int. J. comput. math., 88, 1519-1532, (2011) · Zbl 1220.15009  Tsitsas, N.L.; Alivizatos, E.G.; Kalogeropoulos, G.H., A recursive algorithm for the inversion of matrices with circulant blocks, Applied math. comput., 188, 877-894, (2007) · Zbl 1125.65026  G. Zhao, The improved nonsingularity on the r-circulant matrices in signal processing, in: International Conference On Computer Techology and Development-ICCTD 2009, Kota Kinabalu, pp. 564-567.  W. Zhao, The inverse problem of anti-circulant matrices in signal processing, in: Pacific-Asia Conference on Knowledge Engineering and Software Engineering-KESE 2009, Shenzhen, pp. 47-50.  Zhang, F., Matrix theory: basic results and techniques, (2011), Springer New York, 2nd · Zbl 1229.15002
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.