Circulant $m$-Diagonal Matrices Associated with Chebyshev Polynomials

Abstract

In this study, we deal with an $m$ banded circulant matrix, generally called circulant $m$-diagonal matrix. This special family of circulant matrices arise in many applications such as prediction, time series analysis, spline approximation, difference solution of partial differential equations, and so on. We firstly obtain the statements of eigenvalues and eigenvectors of circulant $m$-diagonal matrix based on the Chebyshev polynomials of the first and second kind. Then we present an efficient formula for the integer powers of this matrix family depending on the polynomials mentioned above. Finally, some illustrative examples are given by using maple software, one of computer algebra systems (CAS).

Keywords

• Chebyshev polynomial
• Circulant matrix
• Eigenvalue
• Eigenvector

References

1. [1] N. L. Tsitsas, E.G. Alivizatos, G.H. Kalogeropoulos, A recursive algorithm for the inversion of matrices with circulant blocks, Appl. Math. Comput., 188(1) (2007), 877-894.
2. [2] G. Zhao, The improved nonsingularity on the r-circulant matrices in signal processing, International Conference On Computer Techology and Development, Kota Kinabalu, (ICCTD 2009), (2009), 564-567.
3. [3] W. Zhao, The inverse problem of anti-circulant matrices in signal processing, Pacific-Asia Conference on Knowledge Engineering and Software Engineering, Shenzhen, (KESE 2009), (2009), 47-50.
4. [4] J. Rimas, Investigation of dynamics of mutually synchronized systems, Telecommun. Radio Eng., 31(2) (1977), 68-79.
5. [5] J. Rimas, G. Leonaite, Investigation of a multidimensional automatic control system with delays and chain form structure, Inf. Technol. Control, 35(1) (2006), 65-70.
6. [6] J. Rimas, On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices–I, Appl. Math. Comput., 165(1) (2005), 137-141.
7. [7] J. Rimas, On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices–II, Appl. Math. Comput., 169(2) (2005), 1016-1027.
8. [8] J. Rimas, On computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices–I, Appl. Math. Comput., 172(1) (2006), 86-90.

9. [9] J. Rimas, On computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices–II, Appl. Math. Comput., 174(1) (2006), 511-552.
10. [10] J. Gutierrez-Gutierrez, Positive integer powers of complex symmetric circulant matrices, Appl. Math. Comput., 202(2) (2008), 877-881.
11. [11] F. Köken, D. Bozkurt, Positive integer powers for one type of odd order circulant matrices, Appl. Math. Comput., 217(9) (2011), 4377-4381. [12] A. Ö teleş, M. Akbulak, A Single Formula for Integer Powers of Certain Real Circulant Matrix of Odd and Even Order, Gen. Math. Notes, 35(2) (2016), 15-28.
12. [13] Z. Jiang, H. Xin, H. Wang, On computing of positive integer powers for r-circulant matrices, Appl. Math. Comput., 265 (2015), 409-413.
13. [14] P. J. Davis, Circulant Matrices, Chelsea Publishing, New York, 1994. [15] R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inform. Theory, 2(3) (2006), 155-239.
14. [16] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, CRC Press, Washington, 2003.
15. [17] C. Kızılateş, N. Tuğlu, B. Çekim, On the (p, q)–Chebyshev Polynomials and Related Polynomials, Mathematics, 7(136) (2019), 1-12.
16. [18] S. Wang, Some new identities of Chebyshev polynomials and their applications, Adv. Differ. Equ., 2015:355 (2015), 1-8.
17. [19] C. Cesarano, Generalized Chebyshev polynomials, Hacet. J. Math. Stat., 43(5) (2014), 731-740.