On Powers, Roots and Moore–Penrose Inverses of Matrices Via Generalized Fibonacci Numbers

Sinan Karakaya; Halim Özdemir; Ahmet Yaşar Özban

doi:10.2478/amsil-2026-0002

.blurhash-client-img { display: none !important; }

On Powers, Roots and Moore–Penrose Inverses of Matrices Via Generalized Fibonacci Numbers

Annales Mathematicae Silesianae

Volume 40 (2026): Issue 2 (September 2026)

By: Sinan Karakaya , Halim Özdemir and Ahmet Yaşar Özban

Open Access

|Feb 2026

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.

References

A. Ben-Israel and T.N.E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, 2003.
Search in Google Scholar Back to article
P. Catarino, H. Campos, and P. Vasco, On the Mersenne sequence, Ann. Math. Inform. 46 (2016), 37–53.
Search in Google Scholar Back to article
T. Goy, On new identities for Mersenne numbers, Appl. Math. E-Notes 18 (2018), 100–105.
Search in Google Scholar Back to article
F.A. Graybill, Matrices with Applications in Statistics, Wadsworth Advanced Books and Software, Belmont, CA, 1983.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
N.J. Higham and L. Lin, On pth roots of stochastic matrices, Linear Algebra Appl. 435 (2011), no. 3, 448–463.
Search in Google Scholar Back to article
D. Kalman and R. Mena, The Fibonacci numbers-exposed, Math. Mag. 76 (2003), no. 3, 167–181.
Search in Google Scholar Back to article
T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
Search in Google Scholar Back to article
H. Özdemir, S. Karakaya, and T. Petik, Notes on generalized Fibonacci numbers and matrices, Honam Math. J. 44 (2022), no. 4, 473–484.
Search in Google Scholar Back to article
P.J. Psarrakos, On the mth roots of a complex matrix, Electron. J. Linear Algebra 9 (2002), 32–41.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, New York, 2000.
Search in Google Scholar Back to article
Z. Şiar and R. Keskin, Some new identities concerning generalized Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 42 (2013), no. 3, 211–222.
Search in Google Scholar Back to article
M.I. Smith, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24 (2003), no. 4, 971–989.
Search in Google Scholar Back to article
S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications, Ellis Horwood Ltd., Chichester, 1989.
Search in Google Scholar Back to article
F.V. Waugh and M.E. Abel, On fractional powers of a matrix, J. Amer. Statist. Assoc. 62 (1967), no. 319, 1018–1021.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2026-0002 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

Journal RSS Feed

Language: English

Page range: 230 - 241

Submitted on: Aug 19, 2025

Accepted on: Feb 1, 2026

Published on: Feb 19, 2026

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Keywords:

11B37,

11B39,

15A09

Related subjects:

Mathematics,

General mathematics

© 2026 Sinan Karakaya, Halim Özdemir, Ahmet Yaşar Özban, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 40 (2026): Issue 2 (September 2026)