On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers
Authors
Q. Fan
- School of Mathematics and Statistics, Linyi University, Linyi 276000, China.
Y. Wei
- School of Mathematics and Statistics, Linyi University, Linyi 276000, China.
Y. Zheng
- School of Automation and Electrical Engineering, Linyi University, Linyi 276000, China.
Z. Jiang
- School of Mathematics and Statistics, Linyi University, Linyi 276000, China.
Abstract
In this paper, we discuss skew circulant matrices involving the product of Pell and Pell-Lucas numbers. The invertibility of the skew circulant matrices is investigated, while the fundamental theorems on the determinants and inverses of such matrices are derived by simple construction matrices. Specifically, the determinant and inverse of \(n\times n\) skew circulant matrices can be expressed by the \((n-1)\)th, \(n\)th, \((n+1)\)th, \((n+2)\)th product of Pell and Pell-Lucas numbers.
Some norms and bounds for spread of these matrices are given, respectively. In addition, we generalized these results to skew left circulant matrix involving the product of Pell and Pell-Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our theoretical results.
Share and Cite
ISRP Style
Q. Fan, Y. Wei, Y. Zheng, Z. Jiang, On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers, Journal of Mathematics and Computer Science, 31 (2023), no. 2, 225--239
AMA Style
Fan Q., Wei Y., Zheng Y., Jiang Z., On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers. J Math Comput SCI-JM. (2023); 31(2):225--239
Chicago/Turabian Style
Fan, Q., Wei, Y., Zheng, Y., Jiang, Z.. "On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers." Journal of Mathematics and Computer Science, 31, no. 2 (2023): 225--239
Keywords
- Determinant
- inverse
- norm
- spread
- Pell number
- skew circulant matrix
MSC
- 15A09
- 15A15
- 15B05
- 11B39
- 65F40
References
-
[1]
K. Balasubramanian, Computational strategies for the generation of equivalence classes of hadamard matrices, J. Chem. Inf. Comput. Sci., 35 (1995), 581–589
-
[2]
E. J. Barbeau, Pell’s Equation, Springer-Verlag, New York (2003)
-
[3]
A. Bose, R. S. Hazra, K. Saha, Spectral norm of circulant-type matrices, J. Theoret. Probab., 24 (2011), 479–516
-
[4]
P. J. Davis, Circulant Matrices, John Wiley & Sons, ew York-Chichester-Brisbane (1979)
-
[5]
Q. Feng, F. Meng, Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method, Optik, 127 (2016), 7450–7458
-
[6]
Y. Gao, Z. Jiang, Y. Gong, On the determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas Numbers, WSEAS Trans. Math., 12 (2013), 472–481
-
[7]
M. V. Houteghem, T. Verstraelen, D. V. Neck, C. Kirschhock, J. A. Martens, M. Waroquier, V. V. Speybroeck, Atomic velocity projection method: A new analysis method for vibrational spectra in terms of internal coordinates for a better understanding of zeolite nanogrowth, J. Chem. Theory Comput., 7 (2011), 1045–1061
-
[8]
A. Ë™Ipek, On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Appl. Math. Comput., 217 (2011), 6011–6012
-
[9]
Z. Jiang, Y. Gong, Y. Gao, Invertibility and explicit inverses of circulant-type matrices with k-Fibonacci and k-Lucas numbers, Abstr. Appl. Anal., 2014 (2014), 9 pages
-
[10]
X. Jiang, K. Hong, Exact determinants of some special circulant matrices involving four kinds of famous numbers, Abstr. Appl. Anal., 2014 (2014), 12 pages
-
[11]
X. Jiang, K. Hong, Explicit inverse matrices of Tribonacci skew circulant type matrices, Appl. Math. Comput., 268 (2015), 93–102
-
[12]
X. Jiang, K. Hong, Explicit form of determinants and inverse matrices of Tribonacci r-circulant type matrices, J. Math. Chem., 56 (2018), 1234–1249
-
[13]
Z. L. Jiang, Z. X. Zhou, Circulant Matrices, Chengdu Technology University Publishing Company, (1999)
-
[14]
H. Karner, J. Schneid, C. W. Ueberhuber, Spectral decomposition of real circulant matrices, Linear Algebra Appl., 367 (2003), 301–311
-
[15]
Q.-X. Kong, J.-T. Jia, A structure-preserving algorithm for linear systems with circulant pentadiagonal coefficient matrices, J. Math. Chem., 53 (2015), 1617–1633
-
[16]
T. Koshy, Pell and Pell-Lucas numbers with applications, Springer, New York (2014)
-
[17]
J. Li, Z. Jiang, F. Lu, Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers, Abstr. Appl. Anal., 2014 (2014), 9 pages
-
[18]
F. Lu , Y. Gong, H. Zhou, The spectrum and spanning trees of polyominos on the torus, J. Math. Chem., 52 (2014), 1841–1847
-
[19]
W. Min, B. P. English, G. Luo, B. J. Cherayil, S. C. Kou, X. S. Xie, Fluctuating enzymes: Lessons from single-molecule studies, Acc. Chem. Res., 38 (2005), 923–931
-
[20]
L. Mirsky, The spread of a matrix, Mathematika, 3 (1956), 127–130
-
[21]
H. J. Nussbaumer, Fast Fourier transform and convolution algorithms, Springer Science & Business Media, (2012)
-
[22]
J. Shao, Z. Zheng, F. Meng, Oscillation criteria for fractional differential equations with mixed nonlinearities, Adv. Differ. Equ., 2013 (2013), 9 pages
-
[23]
R. Sharma, R. Kumar, Remark on upper bounds for the spread of a matrix, Linear Algebra Appl., 438 (2013), 4359–4362
-
[24]
S.-Q. Shen, J.-M. Cen, Y. Hao, On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput., 217 (2011), 9790–9797
-
[25]
W.-C. Siu, A. G. Constantinides, Very fast discrete Fourier transform using number theoretic transform, IEE Proceedings G (Electronic Circuits and Systems), 130 (1983), 201–204
-
[26]
Y. G. Sun, F. W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput., 198 (2008), 375–381
-
[27]
J. Wang, F. Meng, S. Liu, Interval oscillation criteria for second order partial differential systems with delays, J. Comput. Appl. Math., 212 (2008), 397–405
-
[28]
Y. Wei, X. Jiang, Z. Jiang, S. Shon, Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices, Adv. Differ. Equ., 2019 (2019), 11 pages
-
[29]
Y. Wei, X. Jiang, Z. Jiang, S. Shon, On inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners, J. Appl. Anal. Comput., 10 (2020), 178–191
-
[30]
Y. Wei, Y. Zheng, Z. Jiang, S. Shon, A study of determinants and inverses for periodic tridiagonal Toeplitz matrices with perturbed corners involving Mersenne numbers, Mathematics, 7 (2019), 1–11
-
[31]
Y. Wei, Y. Zheng, Z. Jiang, S. Shon, The inverses and eigenpairs of tridiagonal Toeplitz matrices with perturbed rows, J. Appl. Math. Comput., 68 (2022), 623–636
-
[32]
J. Wu, P. Zhang, W. Liao, Upper bounds for the spread of a matrix, Linear Algebra Appl., 437 (2012), 2813–2822
-
[33]
R. Xu, F. Meng, Some new weakly singular integral inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2016 (2016), 16 pages
-
[34]
Y. Yazlik, N. Taskara, On the norms of an r-circulant matrix with the generalized k-Horadam numbers, J. Inequal. Appl., 2013 (2013), 8 pages
-
[35]
D. Yerchuck, A. Dovlatova, Quantum optics effects in quasi-one-dimensional and two-dimensional carbon materials, J. Phys. Chem, 116 (2012), 63–80
-
[36]
Y. Zheng, S. Shon, Exact determinants and inverses of generalized Lucas skew circulant type matrices, Appl. Math. Comput., 270 (2015), 105–113
