On decomposition of k-tridiagonal ℓ-Toeplitz matrices and its applications

Abstract

We consider a k-tridiagonal ℓ-Toeplitz matrix as one of generalizations of a tridiagonal Toeplitz matrix. In the present paper, we provide a decomposition of the matrix under a certain condition. By the decomposition, the matrix is easily analyzed since one only needs to analyze the small matrix obtained from the decomposition. Using the decomposition, eigenpairs and arbitrary integer powers of the matrix are easily shown as applications.

A periodic determinantal property for (0,1) double banded matrices

“…These matrices were extended in many instances. As k-tridiagonal matrices as we can find in [2][3][4], some extensions [5,6] motivated by (1), or in more general non-symmetric patterned cases as in [6][7][8][9]. In fact, they all can be seen as particular cases of the more general family of the banded matrices (cf.…”

On the determinant of general pentadiagonal matrices

“…The first explicit denomination of k-tridiagonal matrices can be found in [15]. We point out that the addition of new sub/super-diagonals equally spaced, as in [2] and [16], is an extension of the original approach proposed by Egerváry and Szász in [4]. A graph theoretical approach can be found in [7], and some extensions in [13] and [18].…”

On determinantal recurrence relations of banded matrices

“…These matrices (particularly, those which are Toeplitz) occur in many numerical problems involving different kinds of differential equations and they also naturally emerge in many areas of pure, applied, and numerical mathematics, engineering, statistics, signal processing, among many others, with particular emphasis to the computational problems related to the calculation of spectra, determinant, permanent, characteristic polynomial, inverse, power, and different types of decompositions (Arbenz, 1991;Arıkan & Kılıc, 2017;Cinkir, 2012;Diele & Lopez, 1998;Elouafi, 2013;Elouafi, 2011;Hadj & Elouafi, 2008;Jia et al, 2016;Kratz, 2010;Kratz & Tentler, 2008;Kılıc & El-Mikkawy, 2008;Montaner & Alfaro, 1995;Marr & Vineyard, 1988;Sweet, 1969). There are many extensions of these matrices which deserved attention in many areas of research as we can find, for example, in (An delić & da Fonseca, 2021;Arbenz, 1991;Egerváry & Szász, 1928;da Fonseca & Yılmaz, 2015;Kratz, 2010;Kratz & Tentler, 2008;Losonczi, 1992;McMillen, 2009;Ohashi et al, 2015;Takahira et al, 2019). All of them belong to the family of the banded matrices.…”