On computing of integer positive powers for one type of tridiagonal and antitridiagonal matrices of even order

Subject Areas : Linear and multilinear algebra; matrix theory

M. Beiranvand ¹ , M. Ghasemi Kamalvand ²

1 - Department of Mathematics, Lorestan University, Khorramabad, Iran
2 - Department of Mathematics, Lorestan University, Khorramabad, Iran

Received: 2021-01-27 Accepted : 2021-03-27 Published : 2021-03-01

Keywords: Eigenvalues, Tridiagonal matrices, eigenvectors, Jordan's form, matrix powers,

Abstract :

In this paper, firstly we derive a general expression for the $m$th power ($m\in\mathbb{N}$) for one type of tridiagonal matrices of even order. Secondly we present a method for computing integer powers of the antitridiagonal matrices that is corresponding with these matrices. Then, we present some examples to illustrate our results and give Maple 18 procedure in order to verify our calculations

References:

[1] R. P. Agarwal, Difference equations and inequalities, Marcel-Dekker, New York. 1992.

[2] M. Beiranvand, M. Ghasemi Kamalvand, Explicit expression for arbitrary positive powers of special tridiagonal matrices, J. Appl. Math. 2020, 2020:7290403.

[3] J. Gutierrez, Binomial cofficients and powers of large tridiagonal matrices with constant diagonals, Appl. Math. Comput. 219 (2013), 9219-9222.

[4] J. Gutierrez, Positive integer powers of certain tridiagonal matrices, Appl. Math. Comput. 202 (2008), 133-140.

[5] P. Horns, Ch. Johnson, Martin Analysis, Cambridge University Press, 1968.

[6] G. James, Advanced Modern Engineering Mathematics, Addisson-Wesley, 1994.

[7] G. Leonaite, J. Rimas, Analysis of multidimensional delay system with chain form structure, Proceeding of the 17th International Symposium on Mathematical Theory of Networks and systems, Kyoto, Japan, 2006.

[8] A. Oteles, M. Akbulak, Positive integer powers of certain complex tidiagonal matrices, Math. Sci. Lett. 2-1 (2013), 63-72.

[9] A. Oteles, M. Akbulak, Positive integer powers of certain complex tidiagonal matrices, Appl. Math. Comput. 219 (2013), 10488-10455.

[10] A. Oteles, M. Akbulak, Positive integer powers of one type of complex tridiagonal matrix, Bull. Malays. Math. Sci. Soc. 37 (4) (2014), 971-981.

[11] R. W. Picard, I. M. Elfadel, Structure of aura and co-occurrence matrices for the Gibbs texture model, J. Math. Imaging. Vision. 2 (1) (1992) 5-25.

[12] J. Rimas, Investigation of dynamics of mutually synchronized system, Telecocommunications. Radio. Engin. 32 (2) (1977), 68-73.

[13] J. Rimas, On Computing of arbitrary positive integer powers for one type of tridiagonal matrices of even order, Appl. Math. Comput. 164 (2005), 829-835.

[14] J. Rimas, On computing of orbitary positive integer powers for one type of tridiagonal matrices with elements 1,0,0,··· ,0,1 in principal and 1,1,1,··· ,1,1 in neighboring diagonals-I, Appl. Math. Comput. 186 (2007), 1254-1257.

[15] J. Rimas, On computing of orbitary positive integer powers for one type of tridiagonal matrices with elements 1,0,0,··· ,0,1 in principal and 1,1,1,··· ,1,1 in neighboring diagonals-II, Appl. Math. Comput. 187 (2007), 1472-1475.

[16] J. Rimas, On computing of orbitary positive integer powers for one type of symmetric tridiagonal matrices of even order, Appl. Math. Comput. 172 (2006), 245-251.

[17] S. E. Umbaugh, Computer Imaging: Digital Image Analysis and Processing, The CRC Pres, 2005.