Cholesky Factorization of the Generalized Symmetric k- Fibonacci Matrix

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. [1] Falcon, S. and Plaza, Á., “The k- Fibonacci sequence and the Pascal 2- triangle”, Chaos, Solitons & Fractals, 33(1): 38-49, (2007).
2. [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. [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. [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. [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. [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. [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. [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. [9] Irmak, N. and Köme, C., “Linear algebra of the Lucas matrix”, Hacettepe Journal of Mathematics and Statistics, 50(2), 549-558, (2021).
10. [10] Kilic, E., “Sums of the squares of terms of sequence u_n”, Proceedings Mathematical Sciences, 118 (1): 27–41, (2008).
11. [11] Koshy, T., “Fibonacci and Lucas Numbers with Applications”, John Wiley & Sons, (2001).