The Linear Algebra of the Pell-Lucas Matrix

Year 2024, , 158 - 168, 30.09.2024

Hayrullah Özimamoğlu , Ahmet Kaya

https://doi.org/10.33401/fujma.1404456

Abstract

In this paper, we introduce the Pell-Lucas and the symmetric Pell-Lucas matrices. The study delves into the linear algebra aspects of these matrices, analyzing their mathematical properties and relationships. We construct decompositions for both the Pell-Lucas matrix and its inverse matrix. We present the Cholesky factorization of the symmetric Pell-Lucas matrices. Furthermore, we derive some valuable identities and bounds for the eigenvalues of these symmetric matrices through the application of majorization notation.

Keywords

Pell-Lucas matrix, symmetric Pell-Lucas matrix, Cholesky factorization, majorization

References

• [1] J. Ercolano, Matrix generators of Pell sequences, Fibonacci Quart., 17(1) (1979), 71-77. $ \href{https://www.fq.math.ca/Scanned/17-1/ercolano.pdf}{\mbox{[Web]}} $
• [2] A.F. Horadam, Pell identities, Fibonacci Quart., 9(3) (1971), 245-252. $\href{https://www.fq.math.ca/Scanned/9-3/horadam-a.pdf}{\mbox{[Web]}} $
• [3] T. Koshy, Pell and Pell-Lucas numbers with applications, New York: Springer, (2014). $ \href{https://link.springer.com/book/10.1007/978-1-4614-8489-9}{\mbox{[Web]}} $
• [4] M. Bicknell, A primer on the Pell sequence and related sequences, Fibonacci Quart., 13(4) (1975), 345-349. $\href{https://www.fq.math.ca/Scanned/13-4/bicknell.pdf}{\mbox{[Web]}} $
• [5] R. Brawer and M. Pirovino, The Linear Algebra of the Pascal Matrix, Linear Algebra Appl., 174 (1992), 13-23. $\href{https://doi.org/10.1016/0024-3795(92)90038-C}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-38249007937&origin=resultslist&sort=plf-f&src=s&sid=26b6f52262b7a43f206286e14ce03f5c&sot=b&sdt=b&s=%28TITLE-ABS-KEY%28the+AND+linear+AND+algebra+AND+of+AND+the+AND+pascal+AND+matrix%29+AND+AUTH%28brawer%29%29&sl=54&sessionSearchId=26b6f52262b7a43f206286e14ce03f5c&relpos=0 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1992JK79500002}{\mbox{[Web of Science]}} $
• [6] G.Y. Lee, J.S. Kim and S.G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart., 40(3) (2002), 203-211. $ \href{http://dx.doi.org/10.1080/00150517.2002.12428645}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0042910454&origin=resultslist&sort=plf-f&src=s&sid=9cb70582b33529f30cd2700d446ff40f&sot=b&sdt=b&s=%28TITLE-ABS-KEY%28Factorizations+and+eigenvalues+of+Fibonacci+and+symmetric+Fibonacci+matrices%29+AND+AUTH%28lee%29%29&sl=60&sessionSearchId=9cb70582b33529f30cd2700d446ff40f&relpos=0 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000176714400002}{\mbox{[Web of Science]}} $
• [7] G.Y. Lee and J.S. Kim, The linear algebra of the k-Fibonacci matrix, Linear Algebra Appl., 373 (2003), 75-87. $ \href{https://doi.org/10.1016/S0024-3795(02)00596-7}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0141569552&origin=resultslist&sort=plf-f&src=s&sid=d681b03ab35358a7b53d35e8d48c6bee&sot=b&sdt=b&s=TITLE-ABS-KEY%28The+linear+algebra+of+the+k-Fibonacci+matrix%29&sl=59&sessionSearchId=d681b03ab35358a7b53d35e8d48c6bee&relpos=4 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000185778400007}{\mbox{[Web of Science]}} $
• [8] Z. Zhang and X.Wang, A factorization of the symmetric Pascal matrix involving the Fibonacci matrix, Discrete Appl. Math., 155(17) (2007), 2371-2376. $ \href{https://doi.org/10.1016/j.dam.2007.06.024}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-34648831937&origin=resultslist&sort=plf-f&src=s&sid=8e1ca21e5472053740f1372ccc332c3a&sot=b&sdt=b&s=%28TITLE-ABS-KEY%28A+factorization+of+the+symmetric+Pascal+matrix+involving+the+Fibonacci+matrix%29+AND+AUTH%28zhang%29%29&sl=55&sessionSearchId=8e1ca21e5472053740f1372ccc332c3a&relpos=0 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000250823100019}{\mbox{[Web of Science]}}$
• [9] Z. Zhang, The linear algebra of the generalized Pascal matrix, Linear Algebra Appl., 250 (1997), 51-60. $\href{https://doi.org/10.1016/0024-3795(95)00452-1}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1997VX45000005}{\mbox{[Web of Science]}} $
• [10] N. Irmak and C. Köme, Linear algebra of the Lucas matrix, Hacettepe J. Math. Stat., 50(2) (2021), 549-558. $ \href{https://doi.org/10.15672/hujms.746184}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85105440972&origin=resultslist&sort=plf-f&src=s&sid=77c9f7d9ec0d3b77c3e5d305a036b280&sot=b&sdt=b&s=TITLE-ABS-KEY%28Linear+algebra+of+the+Lucas+matrix%29&sl=69&sessionSearchId=77c9f7d9ec0d3b77c3e5d305a036b280&relpos=5 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000640069900022}{\mbox{[Web of Science]}} $
• [11] C. Köme, Cholesky factorization of the generalized symmetric k-Fibonacci matrix, Gazi Univ. J. Sci., 35(4) (2022), 1585-1595. $\href{https://doi.org/10.35378/gujs.838411}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85140066069&origin=resultslist&sort=plf-f&src=s&sid=ded2a8a39752061915044770426d360b&sot=b&sdt=b&s=TITLE-ABS-KEY%28Cholesky+factorization+of+the+generalized+symmetric%29&sl=85&sessionSearchId=ded2a8a39752061915044770426d360b&relpos=1 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000904870300024}{\mbox{[Web of Science]}} $
• [12] S. Vasanthi and B. Sivakumar, Jacobsthal matrices and their properties, Indian J. Sci. Tech., 15(5) (2022), 207-215. $\href{https://doi.org/10.17485/IJST/v15i5.1948}{\mbox{[CrossRef]}} $
• [13] E. Kılıç and D. Taşçı, The linear algebra of the Pell matrix, Bol. Soc. Mat. Mex., 11(3) (2005), 163–174. $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-33745641909&origin=resultslist&sort=plf-f&src=s&sid=8b018631a936a5f87927f918d6583095&sot=b&sdt=b&s=TITLE-ABS-KEY%28The+linear+algebra+of+the+Pell+matrix%29&sl=52&sessionSearchId=8b018631a936a5f87927f918d6583095&relpos=6 } {\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000236135900001}{\mbox{[Web of Science]}} $
• [14] A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics, (2011). $ \href{https://doi.org/10.1007/978-0-387-68276-1}{\mbox{[CrossRef]}} $
• [15] D.S. Mitrinovic and P.M. Vasic, Analytic Inequalities (Vol. 1)., Berlin: Springer, (1970).$ \href{https://doi.org/10.1007/978-3-642-99970-3}{\mbox{[CrossRef]}} $
• [16] G.H. Hardy, J. E. Littlewood and G. Polya, Some simple inequalities satisfied by convex functions, Messenger Math., 58 (1929), 145–152.$ $
• [17] A.F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart., 23(1) (1985), 7-20. $ \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1985AED6800002}{\mbox{[Web of Science]}} $