On powers, roots and Moore-Penrose inverses of matrices via generalized Fibonacci numbers

Sinan Karakaya; Halim Özdemir; Ahmet Yaşar Özban

Published: 2026-02-19

2025

On powers, roots and Moore-Penrose inverses of matrices via generalized Fibonacci numbers

Sinan Karakaya

, Halim Özdemir

, Ahmet Yaşar Özban

Abstract

The purpose of this work is to introduce some new results about the relations between powers, roots and Moore–Penrose inverses of square matrices satisfying a cubic matrix equation and the generalized Fibonacci numbers. The results can also be used for rectangular matrices. Moreover, we give some numerical examples to verify theoretical results.

Keywords:

generalized Fibonacci numbers , Generalized Lucas numbers , Moore-Penrose Inverse , Matrix roots

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Karakaya, S., Özdemir, H., & Özban, A. Y. (2026). On powers, roots and Moore-Penrose inverses of matrices via generalized Fibonacci numbers. Annales Mathematicae Silesianae. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/20513

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References

A. Ben-Israel and T.N.E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003.
Google Scholar

P. Catarino, H. Campos, and P. Vasco, On the Mersenne sequence, Ann. Math. Inform. 46 (2016), 37–53.
Google Scholar

T. Goy, On new identities for Mersenne numbers, Appl. Math. E-Notes 18 (2018), 100–105.
Google Scholar

F.A. Graybill, Matrices with Applications in Statistics, Wadsworth Advanced Books and Software, Belmont, CA, 1983.
Google Scholar

C.-H. Guo and N.J. Higham, A Schur-Newton method for the matrix pth root and its inverse, SIAM J. Matrix Anal. Appl. 28 (2006), no. 3, 788–804.
Google Scholar

N.J. Higham and L. Lin, A Schur-Padé algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl. 32 (2011), no. 3, 1056–1078.
Google Scholar

N.J. Higham and L. Lin, On pth roots of stochastic matrices, Linear Algebra Appl. 435 (2011), no. 3, 448–463.
Google Scholar

D. Kalman and R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003), no. 3, 167–181.
Google Scholar

T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
Google Scholar

H. Özdemir, S. Karakaya, and T. Petik, Notes on generalized Fibonacci numbers and matrices, Honam Math. J. 44 (2022), no. 4, 473–484.
Google Scholar

P.J. Psarrakos, On the mth roots of a complex matrix, Electron. J. Linear Algebra 9 (2002), 32–41.
Google Scholar

C.R. Rao and S.K. Mitra, Generalized inverse of a matrix and its applications, in: L.M. Le Cam et al. (eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. I: Theory of Statistics, University of California Press, Berkeley, CA, 1972, pp. 601–620.
Google Scholar

P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, New York, 2000.
Google Scholar

Z. Şiar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (2013), no. 3, 211–222.
Google Scholar

M.I. Smith, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24 (2003), no. 4, 971–989.
Google Scholar

S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications, Ellis Horwood Ltd., Chichester, 1989.
Google Scholar

F.V. Waugh and M.E. Abel, On fractional powers of a matrix, J. Amer. Statist. Assoc. 62 (1967), no. 319, 1018–1021.
Google Scholar

2025
Published: 2025-11-02

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

Licence CC

This work is licensed under a Creative Commons Attribution 4.0 International License.

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