×

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers. (English) Zbl 1222.15010

The authors introduce the circulant matrix of the first Fibonacci numbers and the circulant matrix of the first Lucas numbers. Essentially they show these matrices are invertible and they find their inverse matrices. Then they prove that the determinant of the first matrix can be expressed only using Fibonacci numbers and its inverse can be expressed only by using Fibonacci numbers. The same is true if one changes “Fibonacci” by “Lucas”.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A09 Theory of matrix inversion and generalized inverses
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11C20 Matrices, determinants in number theory
15B36 Matrices of integers
Full Text: DOI

References:

[1] Stanimirović, P.; Nikolov, J.; Stanimirović, I., A generalization of Fibonacci and Lucas matrices, Discrete Appl. Math., 156, 2606-2619 (2008) · Zbl 1188.15024
[2] 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
[3] Zhang, Z.; Zhang, Y., The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math., 38, 5, 457-465 (2007) · Zbl 1149.15014
[4] Miladinović, M.; Stanimirović, P., Singular case of generalized Fibonacci and Lucas matrices, J. Korean Math. Soc., 48, 33-48 (2011) · Zbl 1226.05017
[5] Akbulak, M.; Bozkurt, D., On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacet. J. Math. Stat., 37, 2, 89-95 (2008) · Zbl 1204.15031
[6] Solak, S., On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160, 125-132 (2005) · Zbl 1066.15029
[7] Solak, S., Erratum to “On the norms of circulant matrices with the Fibonacci and Lucas numbers” [Appl. Math. Comput. 160(2005) 125-132], Appl. Math. Comput., 190, 1855-1856 (2007) · Zbl 1130.15015
[8] Shen, S.; Cen, J., On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 216, 2891-2897 (2010) · Zbl 1211.15029
[9] Shen, S.; Cen, J., On the norms of circulant matrices with the (k,h)-Fibonacci and (k,h)-Lucas numbers, Int. J. Contemp. Math. Sciences, 6, 18, 887-894 (2011) · Zbl 1248.15020
[10] Good, I. J., On the inversion of circulant matrices, Biometrika, 37, 185-186 (1950) · Zbl 0037.14502
[11] Davis, P. J., Circulant Matrices (1979), Wiley: Wiley New York · Zbl 0418.15017
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.