Journal of Applied Mathematics

Volume 2014 (2014), Article ID 239693, 10 pages

http://dx.doi.org/10.1155/2014/239693

Research Article

The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

Jin-jiang Yao and Zhao-lin Jiang

Department of Mathematics, Linyi University, Linyi, Shandong 276000, China

Received 6 January 2014; Accepted 17 February 2014; Published 17 April 2014

Academic Editor: Feng Gao

Copyright © 2014 Jin-jiang Yao and Zhao-lin Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

1. Introduction

Circulant and skew-circulant matrices are appearing increasingly often in scientific and engineering applications. Briefly, scanning the recent literature, one can see their utility is appreciated in the design of digital filters [1–3], image processing [4–6], communications [7], signal processing [8], and encoding [9]. They have been put on firm basis with the work of Davis [10] and Jiang and Zhou [11].

The skew circulant matrices as preconditioners for linear multistep formulae- (LMF-) based ordinary differential equations (ODEs) codes. Hermitian and skew-Hermitian Toeplitz systems are considered in [12–15]. Lyness and Sørevik employed a skew circulant matrix to construct -dimensional lattice rules in [16]. Spectral decompositions of skew circulant and skew left circulant matrices were discussed in [17]. Compared with cyclic convolution algorithm, the skew cyclic convolution algorithm [8] is able to perform filtering procedure in approximate half of computational cost for real signals. In [2] two new normal-form realizations are presented which utilize circulant and skew circulant matrices as their state transition matrices. The well-known second-order coupled form is a special case of the skew circulant form. Li et al. [18] gave the style spectral decomposition of skew circulant matrix firstly and then dealt with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix. In [3], a new fast algorithm for optimal design of block digital filters (BDFs) was proposed based on skew circulant matrix.

Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [10, 11]. Unfortunately, the computational complexity of these algorithms is very amazing with the order of matrix increasing. However, some authors gave the explicit determinants and inverse of circulant and skew circulant involving some famous numbers. For example, Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [19]. Lind presented the determinants of circulant and skew circulant involving the Fibonacci numbers [20]. Dazheng [21] gave the determinant of the Fibonacci-Lucas quasicyclic matrices. Shen et al. considered circulant matrices with the Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices [22]. Gao et al. [23] gave explicit determinants and inverses of skew circulant and skew left circulant matrices with the Fibonacci and Lucas numbers. Jiang et al. [24, 25] considered the skew circulant and skew left circulant matrices with the -Fibonacci numbers and the -Lucas numbers and discussed the invertibility of the these matrices and presented their determinant and the inverse matrix by constructing the transformation matrices, respectively.

Recently, there are several papers on the norms of some special matrices. Solak [26] established the lower and upper bounds for the spectral norms of circulant matrices with the classical Fibonacci and Lucas numbers entries. İpek [27] investigated an improved estimation for spectral norms of these matrices. Shen and Cen [28] gave upper and lower bounds for the spectral norms of -circulant matrices in the forms of , , and they also obtained some bounds for the spectral norms of Kronecker and Hadamard products of matrix and matrix . Akbulak and Bozkurt [29] found upper and lower bounds for the spectral norms of Toeplitz matrices such that and . The convergence in probability and in distribution of the spectral norm of scaled Toeplitz, circulant, reverse circulant, symmetric circulant, and a class of -circulant matrices is discussed in [30].

Beginning with Mirsky [31], several authors [32–38] have obtained bounds for the spread of a matrix.

The purpose of this paper is to obtain the explicit determinants, explicit inverses, norm, and spread of skew circulant type matrices involving any continuous Lucas numbers. And we generalize the result [23]. In passing, the norm and spread of skew circulant type matrices have not been researched. It is hoped that this paper will help in changing this. More work continuing the present paper is forthcoming.

In the following, let be a nonnegative integer. We adopt the following two conventions , and, for any sequence , in the case .

The Lucas sequences are defined by the following recurrence relations [21–23, 27–29]:

for . The first few values of the sequences are given by the following table:

The is given by the formula

where and are the roots of the characteristic equation .

Definition 1 (see [17]). A skew circulant matrix over with the first row is meant a square matrix of the form

denoted by .

Definition 2 (see [17]). A skew left circulant matrix over with the first row is meant a square matrix of the form

denoted by .

Lemma 3 (see [10, 17]). Let be skew circulant matrix; then

(i) is invertible if and only if the eigenvalues of

 where , , and ;(ii)if is invertible, then the inverse of is a skew circulant matrix.

Lemma 4 (see [17]). Let be skew left circulant matrix and let n be odd; then

where , are the eigenvalues of .

Lemma 5 (see [23]). With the orthogonal skew left circulant matrix

it holds that

Lemma 6 (see [23]). If

then

Lemma 7 (see [27, 28]). Let be the Lucas numbers; then

Definition 8 (see [29]). Let be an matrix. The maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm of matrix are, respectively,

where denotes the conjugate transpose of .

Lemma 9 (see [30]). If is an real symmetric or normal matrix, then one has

where () are the eigenvalues of .

Definition 10 (see [31, 32]). Let be an matrix with eigenvalues , . The spread of is defined as

Beginning with Mirsky [31], several authors [32–38] have obtained bounds for the spread of a matrix.

Lemma 11. Let be an matrix. An upper bound for the spread due to Mirsky [31] states that

where denotes the Frobenius norm of and is trace of .

Lemma 12 (see [38]). Let be an matrix; then

(i)if is real and normal, then

(ii)and if is Hermitian, then

2. Determinant and Inverse of Skew Circulant Matrix with the Lucas Numbers

In this section, let be skew circulant matrix. Firstly, we give a determinant explicit formula for the matrix . Afterwards, we prove that is an invertible matrix for , and then we find the inverse of the matrix .

In the following, let , , , , and .

Theorem 13. Let be skew circulant matrix; then

where is the th Lucas number. Specially, when , one gets the result of [23].

Proof. Obviously, satisfies the equation. In the case , let

be two matrices; then we have

where

So it holds that

While taking , we have

This completes the proof.

Theorem 14. Let be skew circulant matrix; then is an invertible matrix. Specially, when , one gets the result of [23].

Proof. Taking in, Theorem 13, we have . Hence is invertible. In the case , since , where , , we have

where , . If there exists such that , we obtain , for , and hence it follows that is a real number. Since

it yields that , so we have for . Since is not the root of the equation . We obtain , for any , while

It follows from Lemma 3 that the conclusion holds.

Lemma 15. Let the matrix be of the form

Then the inverse of the matrix is equal to

Specially, when , one gets the result of [23].

Proof. Let . Obviously, for . In the case , we obtain . For , we obtain

Hence, we get , where is identity matrix. Similarly, we can verify that . Thus, the proof is completed.

Theorem 16. Let be skew circulant matrix; then

where

Specially, when , one gets the result of [23].

Proof. Let

where

Then, we have

so , where is a diagonal matrix and is the direct sum of and . If we denote , then we obtain .

Since the last row elements of the matrix are (), then the last row elements of the matrix are ( ), where

Hence, it follows from Lemma 15 that letting , then its last row elements are () which are given by the following equations:

Hence, we obtain

where

This completes the proof.

3. Norm and Spread of Skew Circulant Matrix with the Lucas Numbers

Theorem 17. Let be skew circulant matrix; then three kinds of norms of are given by

Proof. By Definition 8 and (12), we have

By Definition 8 and (13), we have

Thus

Theorem 18. Let

be an odd-order alternative skew circulant matrix and let be odd. Then

Proof. By Lemma 3, we have

So

for all .

Since is odd, is an eigenvalue of ; that is,

To sum up, we have

Since all skew circulant matrices are normal, by Lemma 9 and (12), and (52), we have

which completes the proof.

Theorem 19. Let be skew circulant matrix; then the bounds for the spread of are

Proof. The trace of , . By (18) and (43), we have

Since

by (12) and (14),

By (19), we have

4. Determinant and Inverse of Skew Left Circulant Matrix with the Lucas Numbers

In this section, let be skew left circulant matrix. By using the obtained conclusions in Section 2, we give a determinant explicit formula for the matrix . Afterwards, we prove that is an invertible matrix for any positive interger . The inverse of the matrix is also presented.

According to Lemmas 5 and 6 and Theorems 13, 14, and 16, we can obtain the following theorems.

Theorem 20. Let be skew left circulant matrix; then

where is the th Lucas number.

Theorem 21. Let be skew left circulant matrix; then is an invertible matrix.

Theorem 22. Let be skew left circulant matrix; then

where

5. Norm and Spread of Skew Left Circulant Matrix with the Lucas Numbers

Theorem 23. Let be skew left circulant matrix. Then three kinds of norms of are given by

Proof. Using the method in Theorem 17 similarly, the conclusion is obtained.

Theorem 24. Let

be an odd-order alternative skew left circulant matrix; then

Proof. According to Lemma 4,

for , and

So

By (66) and (67), we have

Since all skew left circulant matrices are symmetrical, by Lemma 9 and (12) and (68), we obtain

Theorem 25. Let be skew left circulant matrix; the bounds for the spread of are

where

Proof. Since is a symmetric matrix, by (20),

The trace of is, if is odd,

By (12), we have

Let ; then, by (18), (62), and (74), we obtain

If is even, then

By (18), (62), and (76), we have

So the result follows.

6. Conclusion

We discuss the invertibility of the skew circulant type matrices with any continuous Lucas numbers and present the determinant and the inverse matrices by constructing the transformation matrices. The four kinds of norms and bounds for the spread of these matrices are given, respectively. In [3], a new fast algorithm for optimal design of block digital filters (BDFs) is proposed based on skew circulant matrix. The reason why we focus our attention on skew circulant is to explore the application of skew circulant in the related field in medicine image, image encryption, and real-time tracking. On the basis of existing application situation [4], we conjecture that SVD decomposition of skew circulant matrix will play an important role in CT-perfusion imaging of human brain. On the basis method of [8] and ideas of [5], we will exploit real-time tracking with kernel matrix of skew circulant structure. A novel chaotic image encryption scheme based on the time-delay Lorenz system is presented in [6] with the description of circulant matrix. We will exploit chaotic image encryption algorithm based on skew circulant operation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of Shandong Province (Grant no. ZR2011FL017), the National Nature Science Foundation of China (Grant no. F020701), and the AMEP of Linyi University, China.

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