آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 131 |
| تعداد مقالات | 1,634 |
| تعداد مشاهده مقاله | 1,899,064 |
| تعداد دریافت فایل اصل مقاله | 1,758,846 |
On norms, spread, characteristic polynomial and determinant of Hankel and Toeplitz matrices with Mersenne sequence | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 13، دوره 11، شماره 3، آذر 2026، صفحه 941-958 اصل مقاله (415.83 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.30037.2287 | ||
| نویسندگان | ||
| Kalika Prasad1؛ Munesh Kumari* 1؛ Jagmohan Tanti2 | ||
| 1Department of Applied Science and Humanities (Mathematics), Government Engineering College Bhojpur, Bihar, India, 802301 | ||
| 2Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, India, 226025 | ||
| چکیده | ||
| In this work, some new properties of the Hankel and Toeplitz matrices are obtained by considering the Mersenne numbers as entries. We developed efficient formulas to compute matrix norms like $\|.\|_1$, $\|.\|_\infty$, Euclidean norm, spread, and the lower and upper bound for the spectral norm of these matrices. Also, the study shows that these matrices are non-singular for $n=2$ and singular for $n\geq 3$. Furthermore, we presented rank, eigenvalues, principal minors, and the characteristic polynomial of them explicitly. | ||
| کلیدواژهها | ||
| Mersenne and Fermat numbers؛ Recursive matrices؛ matrix norms and spread؛ rank؛ characteristic polynomial | ||
| مراجع | ||
|
[1] M. Akbulak and D. Bozkurt, On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacet. J. Math. Stat. 37 (2008), no. 2, 89–95.
[2] S. Aktas and Y. Soykan, A study on the norms of Toeplitz matrices with the generalized mersenne numbers, Arch. Curr. Res. Int. 23 (2023), no. 7, 143–157. https://doi.org/10.9734/acri/2023/v23i7600
[3] Y. Alp and E.G. Kocer, The Gram and Hankel matrices via special number sequences, Honam Math. J. 45 (2023), no. 3, 418–432. https://doi.org/10.5831/HMJ.2023.45.3.418
[4] P. Catarino, H. Campos, and P. Vasco, On the Mersenne sequence, Ann. Math. Inform. 46 (2016), 37–53.
[5] M. Chelgham and A. Boussayoud, On the k-Mersenne–Lucas numbers, Notes Number Theory Discrete Math. 27 (2021), no. 1, 7–13. https://doi.org/10.7546/nntdm.2021.27.1.7-13
[6] A. Daşdemir, On the norms of Toeplitz matrices with the Pell, Pell-Lucas and modified Pell numbers, J. Eng. Technol. Appl. Sci. 1 (2016), no. 2, 51–57. https://doi.org/10.30931/jetas.283838
[7] A.F. Horadam, Chebyshev and Fermat polynomials for diagonal functions, Fibonacci Q. 17 (1979), no. 4, 328–333. https://doi.org/10.1080/00150517.1979.12430205
[8] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 2012.
[9] G.Ö. Kızılırmak, On some identities and Hankel matrices norms involving new defined generalized modified Pell numbers, J. New Results Sci. 10 (2021), no. 3, 60–66. https://doi.org/10.54187/jnrs.989508
[10] M. Kumari, K. Prasad, and H. Mahato, On the norms of Toeplitz and Hankel matrices with balancing and Lucas-balancing numbers, Advanced Mathematical Techniques in Computational and Intelligent Systems, CRC Press, 2023, pp. 196–210.
[11] M. Kumari, K. Prasad, J. Tanti, and E. Özkan, On the properties of $r$-circulant matrices involving Mersenne and Fermat numbers, Int. J. Nonlinear Anal. Appl. 14 (2023), no. 5, 121–131. https://doi.org/10.22075/ijnaa.2023.27875.3742
[12] R. Mathias, The spectral norm of a nonnegative matrix, Linear Algebra Appl. 139 (1990), 269–284. https://doi.org/10.1016/0024-3795(90)90403-Y
[13] L. Mirsky, The spread of a matrix, Mathematika 3 (1956), 127–130. https://doi.org/10.1112/S0025579300001790
[14] B.J. Olson, S.W. Shaw, C. Shi, C. Pierre, and R.G. Parker, Circulant matrices and their application to vibration analysis, Appl. Mech. Rev. 66 (2014), no. 4, 040803. https://doi.org/10.1115/1.4027722
[15] K. Prasad and M. Kumari, Some new properties of Frank matrices with entries Mersenne numbers, Natl. Acad. Sci. Lett. (2024), In press. https://doi.org/10.1007/s40009-024-01538-6
[16] K. Prasad, M. Kumari, and H. Mahato, A modified public key cryptography based on generalized Lucas matrices, Commun. Comb. Optim. (2024), In press. https://doi.org/10.22049/cco.2024.28022.1419
[17] R.M. Robinson, Mersenne and Fermat numbers, Proc. Amer. Math. Soc. 5 (1954), no. 5, 842–846. https://doi.org/10.2307/2031878
[18] N. Saba, A. Boussayoud, and K.V. Kanuri, Mersenne-Lucas numbers and complete homogeneous symmetric functions, J. Math. Comput. Sci. 24 (2021), no. 2, 127–139. http://dx.doi.org/10.22436/jmcs.024.02.04
[19] S. Shen, On the norms of Toeplitz matrices involving k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sciences 7 (2012), no. 8, 363–368.
[20] S. Solak and M. Bahşi, On the spectral norms of Hankel matrices with Fibonacci and Lucas numbers, Selcuk J. Appl. Math. 12 (2011), no. 1, 71–76. http://dx.doi.org/10.12988/ijma.2015.411370
[21] Ş. Uygun, On the bounds for the norms of Toeplitz matrices with the Jacobsthal and Jacobsthal Lucas numbers, J. Eng. Technol. Appl. Sci. 4 (2019), no. 3, 105–114. https://doi.org/10.30931/jetas.569742
[22] P. Vasco, P. Catarino, H. Campos, A.P. Aires, and A. Borges, $k$-Pell, $k$-Pell-Lucas and modified $k$-Pell numbers: Some identities and norms of Hankel matrices, Int. J. Math. Anal. 9 (2015), no. 1, 31–37. http://dx.doi.org/10.12988/ijma.2015.411370
[23] A.C. Wilde, Differential equations involving circulant matrices, Rocky Mt. J. Math. 13 (1983), no. 1, 1–13.
[24] Y. Yazlik, N. Yilmaz, and N. Taskara, On the norms of Hankel matrices with the $k$-Jacobsthal and $k$-Jacobsthal Lucas numbers, J. Selcuk Uni. Natural Appl. Sci. 3 (2014), no. 2, 35–42.
[25] G. Zielke, Some remarks on matrix norms, condition numbers, and error estimates for linear equations, Linear Algebra Appl. 110 (1988), 29–41. https://doi.org/10.1016/0024-3795(83)90130-1 | ||
|
آمار تعداد مشاهده مقاله: 527 تعداد دریافت فایل اصل مقاله: 429 |
||