Cholesky Factorization of the Generalized Symmetric k- Fibonacci Matrix

Year 2022, Volume: 35 Issue: 4, 1585 - 1595, 01.12.2022

Cahit Köme

https://doi.org/10.35378/gujs.838411

Cited By: 1

Abstract

Matrix methods are a useful tool while dealing with many problems stemming from linear recurrence relations. In this paper, we discuss factorizations and inverse factorizations of two kinds of generalized k-Fibonacci matrices. We derive some useful identities of the k-Fibonacci sequence. We investigate the Cholesky factorization of the generalized symmetric k-Fibonacci matrix by using these identities.

Keywords

Generalized k-Fibonacci matrix, Symmetric generalized k-Fibonacci matrix, Cholesky factorization

References

• [1] Falcon, S. and Plaza, Á., “The k- Fibonacci sequence and the Pascal 2- triangle”, Chaos, Solitons & Fractals, 33(1): 38-49, (2007).
• [2] Kilic, E. and Tasci, D., “The linear algebra of the Pell matrix”, Boletín de la Sociedad Matemática Mexicana, 11(3): 163–174, (2005).
• [3] Lee, G. Y., Kim, J. S. and Lee, S. G., “Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices”, Fibonacci Quarterly, 40(3): 203–211, (2002).
• [4] Lee, G.Y. and Kim, J.S., “The linear algebra of the k- Fibonacci matrix”, Linear Algebra and its Applications, 373: 75–87, (2003).
• [5] Zhang, Z. and Zhang, Y., “The Lucas matrix and some combinatorial identities”, Indian Journal of Pure and Applied Mathematics, 38(5): 457–465, (2007).
• [6] Stanica, P., “Cholesky factorizations of matrices associated with r-order recurrent sequences”, Integers: Electronic Journal of Combinatorial Number Theory, 5(2): A16, (2005).
• [7] Kocer, E.G., Mansour, T. and Tuglu, N., “Norms of Circulant and Semicirculant Matrices with the Horadam Numbers”, ARS Combinatoria, 85, 353–359, (2007).
• [8] Kızılates, C. and Tuglu, N., “On the Bounds for the Spectral Norms of Geometric Circulant Matrices”, Journal of Inequalities and Applications, 312, (2016).
• [9] Irmak, N. and Köme, C., “Linear algebra of the Lucas matrix”, Hacettepe Journal of Mathematics and Statistics, 50(2), 549-558, (2021).
• [10] Kilic, E., “Sums of the squares of terms of sequence u_n”, Proceedings Mathematical Sciences, 118 (1): 27–41, (2008).
• [11] Koshy, T., “Fibonacci and Lucas Numbers with Applications”, John Wiley & Sons, (2001).