**Mücahit Akbıyık, Seda Yamaç Akbıyık and Fatih Yılmaz**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 3, Pages 399–410

DOI: 10.7546/nntdm.2022.28.3.399-410

**Full paper (PDF, 209 Kb)**

## Details

### Authors and affiliations

**Mücahit Akbıyık**

*Department of Mathematics, Beykent University
Istanbul, Turkey*

**Seda Yamaç Akbıyık**

*Department of Computer Engineering, Istanbul Gelisim University
Istanbul, Turkey*

**Fatih Yılmaz**

*Department of Mathematics, Ankara Hacı Bayram Veli University
Ankara, Turkey *

### Abstract

This paper focuses on a specially constructed matrix whose entries are harmonic Fibonacci numbers and considers its Hadamard exponential matrix. A lot of admiring algebraic properties are presented for both of them. Some of them are determinant, inverse in usual and in the Hadamard sense, permanents, some norms, etc. Additionally, a MATLAB-R2016a code is given to facilitate the calculations and to further enrich the content.

### Keywords

- Harmonic Fibonacci numbers
- Norm
- Determinant
- Permanent

### 2020 Mathematics Subject Classification

- 11B39
- 15A09
- 15A15
- 65F35

### References

- Bahsi, M., & Solak, S. (2010). On the matrices with Harmonic numbers.
*Gazi Journal of Science*, 23(4), 445–448. - Bapat, R. B. (1988). Multinomial probabilities, permanents and a conjecture of Karlin and Rmou.
*Proceedings of the American Mathematical Society*, 102(3), 467–472. - Bozkurt, D. (2011).
*A note on the spectral norms of the matrices connected integer numbers sequence.*arXiv. Available online at: https://arxiv.org/abs/1105.1724v1 - Brualdi, R. A., Gibson, P.M. (1977). Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function.
*Journal of Combinatorial Theory*, Series A, 22, 194–230. - Graham, R., Knuth, D., & Patashnik, K. (1989).
*Concrete Mathematics*. Addison-Wesley, Reading. - Ipek, A., & Akbulak, M. (2012). Hadamard exponential Hankel Matrix, its eigenvalues and some norms.
*Mathematical Sciences Letters*, 1, 81–87. - Jafari Petrudi, S. H., & Pirouz, B. (2015). A particular matrix, its inversion and some norms.
*Applied and Computational Mathematics*, 4(2), 47–52. - Reams, R. (1999). Hadamard inverses, square roots and products of almost semidefinite matrices.
*Linear Algebra and Its Applications*, 288, 35–43. - Solak, S., & Bahsi, M. (2013). On the spectral norms of Toeplitz matrices with Fibonacci and Lucas numbers, Nonlinear Analysis.
*Hacettepe Journal of Mathematics and Statistics*, 42(1), 15–19. - Tuglu, N., & Kızılates, C. (2015). On the Norms of Some Special Matrices with the Harmonic Fibonacci Numbers.
*Gazi University Journal of Science*, 28(3), 497–501. - Tuglu, N., Kızılates, C., & Kesim, S. (2015). On the harmonic and hyperharmonic Fibonacci numbers.
*Advances in Difference Equations*, 1–12. - Yılmaz, F., & Bozkurt, D. (2011). Hessenberg matrices and the Pell and Perrin numbers.
*Journal of Number Theory*, 131, 1390–1396. - Zhang, F. (1999).
*Matrix Theory: Basic Results and Techniques*. Springer-Verlag, New York.

### Manuscript history

- Received: 17 February 2022
- Revised: 1 July 2022
- Accepted: 8 July 2022
- Online First: 9 July 2022

## Related papers

## Cite this paper

Akbıyık, M., Akbıyık, S. Y., & Yılmaz, F. (2022). On linear algebra of one type of symmetric matrices with harmonic Fibonacci entries. *Notes on Number Theory and Discrete Mathematics*, 28(3), 399-410, DOI: 10.7546/nntdm.2022.28.3.399-410.