Powers of Complex Persymmetric Antitridiagonal Matrices with Constant Antidiagonals

Abstract

We derive a general expression for the pth power of any complex persymmetric antitridiagonal Hankel (constant antidiagonals) matrices. Numerical examples are presented, which show that our results generalize the results in the related literature (Rimas 2008, Wu 2010, and Rimas 2009).

1. Introduction

Solving some difference, differential, and delay differential equations, we meet the necessity to compute the arbitrary positive integer powers of square matrix. Recently, computing the integer powers of antitridiagonal matrices has been a very popular problem. There have been several papers on computing the positive integer powers of various kinds of square matrices by Rimas et al., and others [1–5]. In 2011, the general expression for the entries of the power of complex persymmetric or skew-persymmetric antitridiagonal matrices with constant antidiagonals is presented by Gutiérrez-Gutiérrez [1]. Rimas [2] gave the general expression of the th power for this type of symmetric odd order antitridiagonal matrices () in 2008. In [3, 4] a similar problem is solved for antitridiagonal matrices having zeros in main skew diagonal and units in the neighbouring diagonals. In 2010, the general expression for the entries of the power of odd order antitridiagonal matrices with zeros in main skew diagonal and elements ; in neighbouring diagonals is derived by Rimas [5]. In 2013, Rimas [6] gave the eigenvalue decomposition for real odd order skew-persymmetric antitridiagonal matrices with constant antidiagonals () and derived the general expression for integer powers of such matrices.

In the present paper, we derive a general expression for the th power of any complex persymmetric antitridiagonal matrices with constant antidiagonals (). This novel expression is both an extension of the one obtained by Rimas for the powers of the matrix with (see [2] for the odd case and [5] for the even case) and an extension of the one obtained by Honglin Wu for the powers of the matrix with (see [3] for the even case).

2. Derivation of General Expression

In this present paper, we study the entries of positive integer power of an complex persymmetric antitridiagonal matrix with constant antidiagonals as follows: where ,  .

Consider the following complex Toeplitz tridiagonal matrix:

The next trivial result relates the matrix with and with the backward identity [1]: where is the Kronecker delta.

Lemma 1. Let ,  , and . Then where and .

Proof. We have This completes the proof.

We will find the th power () of the matrix (1). Theorem 2 relates all positive integer powers of the matrix to and .

Theorem 2. If ,  , and and if , then where .

Proof. We will proceed by induction on . The case is obvious.
Suppose that the result is true for and consider case .
By the induction hypothesis we have Since we obtain that Since is symmetric and , we have This completes the proof.

Next, we have to solve .

We begin this work by reviewing a theorem regarding the Hermitian Toeplitz tridiagonal matrix .

Theorem 3. Let ,  , and . Then has eigenvalues

Proof. See [7].

With the tridiagonal matrix , we associate the polynomial sequence characterized by a three-term recurrence relation:

With initial conditions and , we can write the relations (11) in matrix form: where and .

Lemma 4. For , the degree of the polynomial is and and has no common root.

Proof. See [7].

One can show that the characteristic polynomial of is precisely . Hence the eigenvalues of are exactly the roots of .

If are the roots of the polynomial , then it follows from (12) that each is an eigenvalue of the matrix and is a corresponding eigenvector [5, 7, 8]. This observation should be taken into account elsewhere in the paper.

The polynomials verify the well-known Christoffel-Darboux Identity.

Lemma 5. We have

Proof. See [7].

Tending to in formula (13), we get

Since the matrix has distinct eigenvalues , thus, the eigendecomposition of the matrix is where and is the transforming matrix formed by the eigenvectors of . Namely, , where are defined as above.

Lemma 6. If , then

Proof. By using the relations (13) and (14), we obtain where if and if .
This completes the proof.

For , we have .

We get immediately the following.

Theorem 7. Assume that and . Then By using the Cauchy Integral Formula, we can give another expressions of the coefficients as follows: where is a closed curve containing the roots of and no roots of .

Corollary 8. If the matrix is nonsingular with , then By using the Cauchy Integral Formula, we can give other expressions of the coefficients : where is a closed curve containing the roots of and no roots of .

Theorem 9. Assume that and . Then By using the Cauchy Integral Formula, we can give other expressions of the coefficients : where is a closed curve containing the roots of and no roots of .

Proof. From Theorem 2 we get Namely, From Theorem 7 it follows that By using the Cauchy Integral Formula, we can give other expressions of the coefficients : This completes the proof.

Corollary 10. Assume that and . Then By using the Cauchy Integral Formula, we can give other expressions of the coefficients : where is a closed curve containing the roots of and no roots of .

3. Numerical Examples

Consider the order antitridiagonal matrix of the following type: Assume that where and are matrix. The polynomial sequence verifies with initial conditions and .

By simple calculation we can show that where are the Chebyshev polynomials [8] of the second kind which satisfies the three-term recurrence relations: with initial conditions and .

Each satisfies and thus the roots of are , . Then, the eigenvalues of are

We get by Theorem 7 the following.

Assume that and . Then

If the matrix is nonsingular and , then

We can obtain the following:

Theorem 11. Consider an odd natural number , . Let and for every . Then for all and , , where are the eigenvalues of the matrix and is the th degree Chebyshev polynomial of the second kind.

Theorem 12. Consider an even natural number , . Let and for every . Then for all and , , where For even order matrix B the following condition is fulfilled: . This means that even order matrix is nonsingular (its determinant is not equal to zero) and derived expression of can be applied for computing negative integer powers, as well. Taking , we get the following expression for elements of the inverse matrix :

Theorem 13. Consider an even natural number , . Let and , . Then

4. Conclusion and Discussion

In this paper, we derive a general expression for the th power of any complex persymmetric antitridiagonal Hankel (constant antidiagonals) matrices with constant antidiagonals (). This novel expression is both an extension of the one obtained by Rimas for the powers of the matrix with (see [2] for the odd case and [5] for the even case) and an extension of the one obtained by Honglin Wu for the powers of the matrix with (see [3] for the even case). We may safely draw the conclusion that our results generalize the results in the related literature [2, 3, 5].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is indebted to the referee for various helpful comments in this paper.