On the Properties of  r-Circulant Matrices Involving Generalized Fermat Numbers

Kuloğlu, Bahar; eser, engin; Özkan, Engin

doi:10.16984/saufenbilder.1280572

On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers

Bahar KULOǦLU, (Milli Eğitim Bakanlığı, Hatuniye Hatuniye Mesleki ve Teknik Anadolu Lisesi, Mardin, Türkiye)

Engin ESER, (Erzincan Binali Yıldırım Üniversitesi, Fen Bilimleri Enstitüsü, Erzincan, Türkiye)

Engin ÖZKAN (Erzincan Binali Yıldırım Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, Erzincan, Türkiye)

Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi

3 0

Yıl: 2023 Cilt: 27 Sayı: 5 Sayfa Aralığı: 956 - 964 Metin Dili: İngilizce DOI: 10.16984/saufenbilder.1280572 İndeks Tarihi: 26-10-2023

On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers

Öz:

r-circulant matrices have applied in numerical computation, signal processing, coding theory, etc. In this study, our main goal is to investigate the r-circulant matrices of generalized Fermat numbers which are shown by We obtain the eigenvalues, determinants, sum identity of matrices. Also we find upper and lower bounds for the spectral norms of generalized Fermat r-circulant matrices. Beside these, we present 𝐺𝑅𝑎,𝑏,𝑟 ∗ matrix in the form of the Hadamard product of two matrices as 𝐺𝑅𝑎,𝑏,𝑟 ∗ = 𝐴. 𝐵. In addition, we get the right and skew- right circulant matrices for . Finally, we examine their different norms (Spectral and Euclidean) and limits for matrix norms.

Anahtar Kelime: Fermat number generalized Fermat numbers norm (spectral and Euclidean) eigenvalues circulant matrices

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] W. Weisstein, Eric, "Proth Number", mathworld.wolfram.com, 2019.
[2] N. J. A. Sloane, “A Handbook of Integer Sequences”, Academic Press, New York, https://oeis.org, 1973.
[3] Generalized Fermat numbers, OEIS. Generalized Fermat numbers - OeisWiki, 2022.
[4] B. Fischer and J. Modersitzki, “Fast inversion of matrices arising in image processing”, Number Algorithms vol. 22 no.1, pp.1-11, 1999.
[5] S. Georgiou and C. Kravvaritis, New good quasi-cyclic codes over GF(3), International Journal of Algebra vol.1, no.1, pp.11-24, 2007.
[6] I. Kra and S. R. Simanca, On circulant matrices, Notices AMS vol.59, no.3, pp.368-377, 2012.
[7] A. C. F. Bueno, “On the Eigenvalues and the Determinant of the Right Circulant Matrices with Pell and Pell– Lucas Numbers”, International Journal of Mathematics and Scientific Computing, vol.4, no.1, pp.19-20, 2014.
[8] S. Solak, “On the norms of circulant matrices with the Fibonacci and Lucas numbers”, Applied Mathematics and Computation, vol.160, no.1, pp.125- 132, 2005.
[9] S. Shen, J. Cen, “On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers”, Applied Mathematics and Computation, vol.216, no.10, pp. 2891- 2897, 2010.
[10] S. Q. Shen, J. M. Cen, “On the spectral norms of r-circulant matrices with the k-Fibonacci and k-Lucas numbers”, International Journal of Contemporary Mathematical Sciences,vol. 5, no. 12, pp. 569-578, 2010.
[11] Y. Zheng, S. Shon, “Exact inverse matrices of Fermat and Mersenne circulant matrix”, In Abstract and Applied Analysis, Hindawi, 2015.
[12] D. Bozkurt, T. Y. Tam, “Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal–Lucas numbers”, Applied Mathematics and Computation, vol.219, no. 2, pp.544- 551, 2012.
[13] M. Kumaria, K. Prasada, J. Tantib, and E. Özkan, “On the properties of г- circulant matrices involving Mersenne and Fermat numbers”, International Journal of Nonlinear Analysis and Applications, vol.1, no.11, 2023.
[14] M. Marin, “Generalized solutions in elasticity of micropolar bodies with voids”, Revista de la Academia Canaria de Ciencias, vol.8, no.1, pp. 101-106, 1996
[15] M. Marin, “Contributions on uniqueness in thermoelastodynamics on bodies with voids”, Ciencias matemáticas (Havana), vol.16, no.2, pp.101-109, 1998.
[16] Z. Pucanovic, and M. Pesovic “Chebyshev polynomials and r- circulant matrices”, Applied Mathematics and Computation, vol.437, no.127521, 2023.
[17] J. M. Blackledget, “Vector and Matrix norm”, Digital Signal Processing, pp.208-236, 2006.
[18] Simon Foucart, http://www.math.drexel.edu/foucart/T eachingFiles/F12/M504Lect6.pdf
[19] R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices”, Linear Algebra and its Application vol.8, no.1, pp.25-33, 1974.
[20] K. Irwin, R. Santiago, Simanca, “On circulant matrices”, ://www.math.colombia.edu/ums/pdf/ci r-not5.pdf

APA	Kuloğlu B, eser e, Özkan E (2023). On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. , 956 - 964. 10.16984/saufenbilder.1280572
Chicago	Kuloğlu Bahar,eser engin,Özkan Engin On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. (2023): 956 - 964. 10.16984/saufenbilder.1280572
MLA	Kuloğlu Bahar,eser engin,Özkan Engin On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. , 2023, ss.956 - 964. 10.16984/saufenbilder.1280572
AMA	Kuloğlu B,eser e,Özkan E On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. . 2023; 956 - 964. 10.16984/saufenbilder.1280572
Vancouver	Kuloğlu B,eser e,Özkan E On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. . 2023; 956 - 964. 10.16984/saufenbilder.1280572
IEEE	Kuloğlu B,eser e,Özkan E "On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers." , ss.956 - 964, 2023. 10.16984/saufenbilder.1280572
ISNAD	Kuloğlu, Bahar vd. "On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers". (2023), 956-964. https://doi.org/10.16984/saufenbilder.1280572

APA	Kuloğlu B, eser e, Özkan E (2023). On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 27(5), 956 - 964. 10.16984/saufenbilder.1280572
Chicago	Kuloğlu Bahar,eser engin,Özkan Engin On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27, no.5 (2023): 956 - 964. 10.16984/saufenbilder.1280572
MLA	Kuloğlu Bahar,eser engin,Özkan Engin On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol.27, no.5, 2023, ss.956 - 964. 10.16984/saufenbilder.1280572
AMA	Kuloğlu B,eser e,Özkan E On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2023; 27(5): 956 - 964. 10.16984/saufenbilder.1280572
Vancouver	Kuloğlu B,eser e,Özkan E On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers. Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2023; 27(5): 956 - 964. 10.16984/saufenbilder.1280572
IEEE	Kuloğlu B,eser e,Özkan E "On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers." Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 27, ss.956 - 964, 2023. 10.16984/saufenbilder.1280572
ISNAD	Kuloğlu, Bahar vd. "On the Properties of r-Circulant Matrices Involving Generalized Fermat Numbers". Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27/5 (2023), 956-964. https://doi.org/10.16984/saufenbilder.1280572