Binomial Coefficients and Powers of One Type of Large Pentadiagonal Matrices

Ahmet Öteleş, Zekeriya Yalcin Karatas

Abstract


In this paper, we derive a general expression for the entries of the $r$th
power $\left( r\in \mathbb{N} \right) $ of one type of the $n\times n$ complex pentadiagonal matrix for all $n \geq 4( r-1)$, in terms of binomial coefficients.


Keywords


Pentadiagonal matrix, Powers of matrices, Binomial coecients, Eigenvalues

Full Text:

PDF

References


R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker (2000), DOI: 10.1201/9781420027020.

S. Arslan, F. Köken and D. Bozkurt, Positive integer powers and inverse for one type of even order symmetric pentadiagonal matrices, Appl. Math. Comput. 219 (2013), 5241 – 5248, DOI: 10.1016/j.amc.2012.11.040.

S. M. Cobb, On powers of matrices with elements in the field of integers modulo 2, Math. Gaz. 42 (342) (1958), 267 – 271, DOI: 10.2307/3610436.

P. M. Crespo and J. Gutiérrez-Gutiérrez, On the elementwise convergence of continuous functions of Hermitian banded Toeplitz matrices, IEEE Transactions on Information Theory 53(3) (2007), 1168 – 1176, DOI: 10.1109/TIT.2006.890697.

J. Feng, A note on computing of positive integer powers for criculant matrices, Appl. Math. Comput. 223 (2013), 472 – 475, DOI: 10.1016/j.amc.2013.08.016.

R. Gray, On the asymptotic eigenvalue distribution of Toeplitz matrices, IEEE Transactions on Information Theory 18 (1972), 725 – 730, DOI: 10.1109/TIT.1972.1054924.

J. Gutiérrez-Gutiérrez, Binomial cofficients and powers of large tridiagonal matrices with constant diagonals, Appl. Math. Comput. 219 (2013), 9219 – 9222, DOI: 10.1016/j.amc.2013.03.090.

J. Gutiérrez-Gutiérrez, Powers of tridiagonal matrices with constant diagonals, Appl. Math. Comput. 206 (2008), 885 – 891, DOI: 10.1016/j.amc.2008.10.005.

J. Rimas and G. Leonaite, Investigation of a multidimensional automatic control system with delays and chain form structure, Inf. Technol. Control 35(1) (2006), 65 – 70, DOI: 10.5755/j01.itc.35.1.12038.

J. Rimas, Investigation of dynamics of mutually synchronized systems, Telecommunications and Radio Engineering 32 (1977) 68 – 79.

J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric pentadiagonal matrices of even order, Appl. Math. Comput. 203 (2008), 582 – 591, DOI: 10.1016/j.amc.2008.04.058.

J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric pentadiagonal matrices of odd order, Appl. Math. Comput. 204 (2008), 120 – 129, DOI: 10.1016/j.amc.2008.06.009.

D. K. Salkuyeh, Positive integer powers of the tridiagonal toeplitz matrices, International Mathematical Forum 1(22) (2006), 1061 – 1065, DOI: 10.12988/imf.2006.06086.




DOI: http://dx.doi.org/10.26713%2Fjims.v11i2.653

eISSN 0975-5748; pISSN 0974-875X