$k-$Circulant Matrices with Generalized Guglielmo Components

Bahadır Yılmaz; Yüksel Soykan

$k-$Circulant Matrices with Generalized Guglielmo Components

Abstract

Matrix theory holds significant importance in addressing real-world problems and computer processes, particularly in the realm of machine learning. The study of special matrices and their linear algebraic properties plays a crucial role in matrix theory, with $k-$circulant matrices standing out as one such special matrix. In this paper, we derive explicit expressions for the sum of elements, the norm associated with the maximum column sum matrix, the norm related to the maximum row sum matrix, the Euclidean norm, eigenvalues, and determinant of a $k-$circulant matrix incorporating the generalized Gulielmo numbers. Additionally, we delve into an examination of the spectral norm pertaining to this $k-$circulant matrix.

Keywords

Generalized Guglielmo numbers, Circulant matrix, $k-$Circulant matrix, Norm, Frobenius (or Euclidean) norm, Spectral norm, Eigenvalues, Determinant

References

[1] Y. Soykan, Generalized Guglielmo numbers: An investigation of properties of triangular, oblong, and pentagonal numbers via their third-order linear recurrence relations, Earthline J. Math. Sci., 9(1) (2022), 1–39.
[2] Y. Soykan, Generalized Tribonacci numbers: Summing formulas, Int. J. Adv. Appl. Math. Mech., 7(3) (2020), 57–76.
[3] B. Radičić, On $k$−circulant matrices (with geometric sequence), Quaest. Math., 39(1) (2016), 135–144. https://doi.org/10.2989/16073606.2015.1024185
[4] P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, 1979.
[5] D. T. Pillai B. J. Chathely A study on circulant matrices and its application in solving polynomial equations, Int. J. Math. Trends Technol., 66(6) (2020), 275–283. https://doi.org/10.14445/22315373/IJMTT-V66I6P527
[6] R. M. Gray, Toeplitz and circulant matrices: A review, Stanford Univ., 2006.
[7] R. E. Cline, R. J. Plemmons, G. Worm, Generalized inverses of certain Toeplitz matrices, Linear Algebra Appl., 8 (1974), 25–33.
[8] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 2012.
[9] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.
[10] C. Kızılateş, N. Tuğlu, On the bounds for the spectral norms of geometric circulant matrices, J. Inequal. Appl., 2016 (2016), Article ID 312. https://doi.org/10.1186/s13660-016-1255-1

[11] E. Polatlı, On the bounds for the spectral norms of $r$−circulant matrices with a type of Catalan triangle numbers, J. Sci. Arts, 3(48) (2019), 575–578.
[12] G. Zielke, Some remarks on matrix norms, condition numbers, and error estimates for linear equations, Linear Algebra Appl., 110 (1988), 29–41.
[13] E. G. Alptekin, Pell, Pell–Lucas ve modified Pell sayıları ile tanımlı circulant ve semicirculant matrisler, Ph.D. Thesis, Selçuk University, 2005.
[14] N. Tuğlu, C. Kızılateş, On the norms of circulant and $r$−circulant matrices with the hyperharmonic Fibonacci numbers, J. Inequal. Appl., 2015 (2015), Article ID 253. https://doi.org/10.1186/s13660-015-0778-1
[15] O. Deveci, E. Karaduman, C. M. Campbell, The Fibonacci-circulant sequences and their applications, Iran. J. Sci. Technol. Trans. A Sci., 41(A4) (2007), 1033–1038.
[16] O. Deveci, On the Fibonacci-circulant $p$−sequences, Util. Math., 108 (2018), 107–124.
[17] R. Turkmen, H. Gökbaş, On the spectral norm of r−circulant matrices with the Pell and Pell–Lucas numbers, J. Inequal. Appl., 2016 (2016), Article ID 65. https://doi.org/10.1186/s13660-016-0997-0
[18] S. Uygun, Some bounds for the norms of circulant matrices with the $k$−Jacobsthal and $k$−Jacobsthal Lucas numbers, J. Math. Res., 8(6) (2016), 133–138.
[19] S. Uygun, S. Yaşamalı, On the bounds for the norms of circulant matrices with the Jacobsthal and Jacobsthal–Lucas numbers, Notes Number Theory Discrete Math., 23(1) (2017), 91–98.
[20] S. Uygun, S. Yaşamalı, On the bounds for the norms of $r$−circulant matrices with the Jacobsthal and Jacobsthal–Lucas numbers, Int. J. Pure Appl. Math., 112(1) (2017), 93–102.
[21] B. Radičić, On k−circulant matrices involving the Jacobsthal numbers, Rev. Un. Mat. Argentina, 60(2) (2019), 431–442.
[22] A. Coşkun, N. Taşkara, On some properties of circulant matrices with third order linear recurrent sequences, Math. Sci. Appl. E-Notes, 6(1) (2018), 12–18.
[23] C. Kızılateş, N. Tuğlu, On the norms of geometric and symmetric geometric circulant matrices with the Tribonacci number, Gazi Univ. J. Sci., 31(2) (2018), 555–567.
[24] A. Pacheenburawana, W. Sintunavarat, On the spectral norms of $r$−circulant matrices with the Padovan and Perrin sequences, J. Math. Anal., 9(3) (2018), 110–122.
[25] W. Sintunavarat, The upper bound estimation for the spectral norm of $r$−circulant and symmetric $r$−circulant matrices with the Padovan sequence, J. Nonlinear Sci. Appl., 9(1) (2016), 92–101. http://dx.doi.org/10.22436/jnsa.009.01.09
[26] Z. Raza, M. Riaz, M. A. Ali, Some inequalities on the norms of special matrices with generalized Tribonacci and generalized Pell–Padovan sequences, arXiv preprint (2015).
[27] Z. Raza, M. A. Ali, On the norms of circulant, $r$−circulant, semi-circulant and Hankel matrices with Tribonacci sequence, arXiv preprint (2014).
[28] Y. Soykan, A study on generalized $(r,s,t)-$−numbers, MathLAB J., 7 (2020), 101–129.
[29] Y. Soykan, On generalized third-order Pell numbers, Asian J. Adv. Res. Rep., 6(1) (2019), 1–18.
[30] Y. Soykan, A closed formula for the sums of squares of generalized Tribonacci numbers, J. Prog. Res. Math., 16(2) (2020), 2932–2941.
[31] Y. Soykan, On the sums of squares of generalized Tribonacci numbers: Closed formulas of $\sum_{k=0}^{n}x^{k}W_{k}^{2}$, Arch. Curr. Res. Int., 20(4) (2020), 22–47. https://doi.org/10.9734/ACRI/2020/v20i430187
[32] Y. Soykan, Summing formulas for generalized Tribonacci numbers, Univ. J. Math. Appl., 3(1) (2020), 1–11. https://doi.org/10.32323/ujma.637876
[33] Y. Soykan, A study on the sums of squares of generalized Tribonacci numbers: Closed form formulas of $\sum_{k=0}^{n}kx^{k}W_{k}^{2}$, J. Sci. Perspect., 5(1) (2021), 1–23. https://doi.org/10.26900/jsp.5.1.02
[34] Y. Soykan, study on sum formulas for generalized Tribonacci numbers: Closed forms of $\sum_{k=0}^{n}kx^{k}W_{k}$ and $\sum_{k=1}^{n}kx^{k}W_{-k}$, J. Prog. Res. Math., 17(1) (2020), 1–40.
[35] Y. Soykan, Formulae for the sums of squares of generalized Tribonacci numbers: Closed form formulas of $\sum_{k=0}^{n}kW_{k}^{2}$, IOSR J. Math., 16(4) (2020), 1–18. https://doi.org/10.9790/5728-1604010118
[36] Y. Soykan, On sum formulas for generalized Tribonacci sequence, J. Sci. Res. Rep., 26(7) (2020), 27–52. https://doi.org/10.9734/JSRR/2020/v26i730283
[37] A. Özkoç, E. Ardıyok, Circulant and negacyclic matrices via Tetranacci numbers, Honam Math. J., 38(4) (2016), 725–738. http://dx.doi.org/10.5831/HMJ. 2016.38.4.725
[38] I. Kra, S. Simanca, On circulant matrices, Notices Amer. Math. Soc., 59(3) (2012), 369–377.

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Bahadır Yılmaz ^*
0000-0002-4685-7101
Türkiye

Yüksel Soykan
0000-0002-1895-211X
Türkiye

Early Pub Date

December 29, 2025

Publication Date

December 29, 2025

Submission Date

July 11, 2025

Acceptance Date

December 27, 2025

Published in Issue

Year 2025 Volume: 1 Number: 1

IZ

https://izlik.org/JA22ER22BB

APA

Yılmaz, B., & Soykan, Y. (2025). $k-$Circulant Matrices with Generalized Guglielmo Components. Sakarya Journal of Mathematics, 1(1), 37-51. https://izlik.org/JA22ER22BB

AMA

1.Yılmaz B, Soykan Y. $k-$Circulant Matrices with Generalized Guglielmo Components. Sakarya Journal of Mathematics. 2025;1(1):37-51. https://izlik.org/JA22ER22BB

Chicago

Yılmaz, Bahadır, and Yüksel Soykan. 2025. “$k-$Circulant Matrices With Generalized Guglielmo Components”. Sakarya Journal of Mathematics 1 (1): 37-51. https://izlik.org/JA22ER22BB.

EndNote

Yılmaz B, Soykan Y (December 1, 2025) $k-$Circulant Matrices with Generalized Guglielmo Components. Sakarya Journal of Mathematics 1 1 37–51.

IEEE

[1]B. Yılmaz and Y. Soykan, “$k-$Circulant Matrices with Generalized Guglielmo Components”, Sakarya Journal of Mathematics, vol. 1, no. 1, pp. 37–51, Dec. 2025, [Online]. Available: https://izlik.org/JA22ER22BB

ISNAD

Yılmaz, Bahadır - Soykan, Yüksel. “$k-$Circulant Matrices With Generalized Guglielmo Components”. Sakarya Journal of Mathematics 1/1 (December 1, 2025): 37-51. https://izlik.org/JA22ER22BB.

JAMA

1.Yılmaz B, Soykan Y. $k-$Circulant Matrices with Generalized Guglielmo Components. Sakarya Journal of Mathematics. 2025;1:37–51.

MLA

Yılmaz, Bahadır, and Yüksel Soykan. “$k-$Circulant Matrices With Generalized Guglielmo Components”. Sakarya Journal of Mathematics, vol. 1, no. 1, Dec. 2025, pp. 37-51, https://izlik.org/JA22ER22BB.

Vancouver

1.Bahadır Yılmaz, Yüksel Soykan. $k-$Circulant Matrices with Generalized Guglielmo Components. Sakarya Journal of Mathematics [Internet]. 2025 Dec. 1;1(1):37-51. Available from: https://izlik.org/JA22ER22BB