Abstract
Tridiagonal matrices appear frequently in mathematical models. In this paper, we derive the positive integer powers of special tridiagonal matrices of arbitrary order.
1. Introduction
Recently, computing the integer powers of tridiagonal matrices has been a very popular problem. Tridiagonal matrices are used in different areas of science and engineering, for instance, the solution of difference systems [1], numerical solution of PDEs [2], telecommunication system analysis [3, 4], texture modeling [5], and image processing and coding [6]. In these areas, the computation of the powers of these matrices is necessary. Therefore, there are a lot of studies dealing with the powers of these matrices using the well-known expression where is Jordan’s form of the matrix and is the transforming matrix. We need the eigenvalues and eigenvectors of these matrices to calculate .
Rimas [7–13], investigated positive integer powers of certain tridiagonal matrices, Oteles and Akbulak [14, 15] and Gutierrez [16] generalized some papers of Rimas, and Wang [17] derived the entries of positive integer powers of complex persymmetric anti-tridiagonal matrices with constant anti-diagonals. Some authors also investigated integer powers of certain tridiagonal matrices.
In this paper, we consider the -th order tridiagonal matrix of the following type where , , , , and are the numbers in the complex . There are many mathematical models that are involved in this form [18].
2. Main Results
According to the following lemmas, Wen-Chyuan Yueh obtains eigenvalue and corresponding eigenvectors for matrix (1), in special cases. In this section, we firstly find the transforming matrices and their invers for the matrix (1). Secondly, we present a general expression for the entries of for .
Lemma 1 (see [19]). Suppose , , and , then the eigenvalues of are given by the corresponding eigenvectors , are given by
In this paper, we need the following theorem.
Theorem 2. (canonical Jordan’s form [20]). Let be any square matrix. Then, there exists a nonsingular matrix which transforms into a block diagonal matrix such that which is called the canonical Jordan’s form, being the eigenvalues of and a Jordan block of the form
Since all the eigenvalues for are distinct ( is a strictly decreasing function of on (0,), and ), columns of the transforming matrix are the eigenvectors of the matrix (1). Also, all eigenvalues correspond to the single Jordan cell in the matrix , then we write down Jordan’s form of the matrix as
From (3), we can write the column transforming matrix as for .
Hence,
Let in which for .
Therefore, we have
So an explicit expression for will suffice.
Theorem 3. Suppose is defined as above, then
Proof. We show that
From (10), equals and follows
If ,
If ,
If and are even or odd, then and are even; therefore, , so we have
If one of or is even and the other is odd, then and are odd; therefore, we have
From (18), (20), and (21) follows
For derivation of the formula for the entries of from (4), (6), (10), (12), and (13), we can conclude
By substituting and in the latter equation and doing the necessary computation follows for , where
Lemma 4 (see [19]). Suppose and and , then the eigenvalues of are given by
The corresponding eigenvectors, , are given by
From (26) for we can write the columns transforming matrix as
Hence,
Let in which for
Therefore, we have
So an explicit expression for will suffice.
Theorem 5. Suppose is defined as above, then
Proof. We show that
From (30) equals and follows
If ,
If ,
If and are even or odd, then is even and is odd; therefore,
So we have
If one of or is even and the other is odd, then and are even; therefore, we have and
Therefore,
Similar for (24), we can conclude for .
Lemma 6 (see [18]). Suppose and , then the eigenvalues of are given by
The corresponding eigenvectors are given by
In the case , the eigenvalues are given by (44) and the corresponding eigenvectors by
In these cases which are similar to previous cases, we showed that integer powers of the matrix for respectively, where , and where
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.