Linear algebra of the Lucas matrix

Abstract

In this paper, we give the factorizations of the Lucas and inverse Lucas matrices. We also investigate the Cholesky factorization of the symmetric Lucas matrix. Moreover, we obtain the upper and lower bounds for the eigenvalues of the symmetric Lucas matrix by using some majorization techniques.

Keywords

• Lucas Matrix
• Cholesky factorization
• eigenvalues
• majorization

References

1. [1] C.M. Fonseca and E. Kılıç, An observation on the determinant of a Sylvester-Kac type matrix, An. Univ. "Ovidius" Constanta Ser. Mat. 28 (1), 111–115, 2020.
2. [2] C.M. Fonseca and Kılıç, A new type of Sylvester–Kac matrix and its spectrum, https://doi.org/10.1080/03081087.2019.1620673.
3. [3] G.H. Hardy, J.E. Littlewood and G. Pólya, Some simple inequalities satisfied by con- vex functions, Messenger Math. 58, 145–152, 1929.
4. [4] C.R. Johnson and R.A. Horn, Matrix analysis, Cambridge University Press Cam- bridge, 1985.
5. [5] E. Kilic and D. Tasci, The linear algebra of the Pell matrix, Bol. Soc. Mat. Mex. 11 (3), 163–174, 2005.
6. [6] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001.
7. [7] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart. 40 (3), 203–211, 2002.
8. [8] G.-Y. Lee and J.-S. Kim, The linear algebra of the k−Fibonacci matrix, Linear Algebra Appl. 373, 75–87, 2003.

9. [9] A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: theory of majorization and its applications 143, Springer, 1979.
10. [10] P. Stanica, Cholesky factorizations of matrices associated with r−order recurrent se- quences, Integers, 5 (2), A16, 2005.
11. [11] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math. 38 (5), 457–465, 2007.