Abstract
We first give the block style spectral decomposition of arbitrary block skew circulant matrix with skew circulant blocks. Secondly, we obtain the singular value of block skew circulant matrix with skew circulant blocks as well. Finally, based on the block style spectral decomposition, we deal with the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks.
1. Introduction
A block skew circulant matrix with skew circulant blocks with the first row is meant a square matrix of the following form: and for any , denoted by . Note that in this paper all facts are based on real field.
Skew circulant matrices have important applications in various disciplines including image processing, signal processing, solving Toeplitz matrix problems, preconditioner, and solving least squares problems in [1–10].
Liu and Guo [11] gave the optimal backward perturbation analysis for a block circulant linear system. Li et al. [12] gave the style spectral decomposition of skew circulant matrix firstly and then dealt with the optimal backward perturbation analysis for the skew circulant linear system. The optimal backward perturbation bounds for underdetermined systems are studied by J.G. Sun and Z. Sun in [13]. Some new theorems generalizing a result of Oettli and Prager are applied to a posteriori analysis of the compatibility of a computed solution to the uncertain data of a linear system by Rigal and Gaches in [14]. Systems with BC structure appear in the context of multichannel signal estimation [15, 16], image restoration [17], cyclic convolution filter banks [18], texture synthesis and recognition [19], and so on.
The block skew circulant matrix with skew circulant blocks is an extension of skew circulant matrix and we believe the block skew circulant linear system with skew circulant blocks can be used in those fields as well. In this paper, firstly, by using the style spectral decomposition of a special skew circulant matrix in [12], we get the block style spectral decomposition of arbitrary block skew circulant matrix with skew circulant blocks. Secondly, we obtain the singular value of block skew circulant matrix with skew circulant blocks as well. Finally, we deal with the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks on the base of its block style spectral decomposition.
2. The Block Style Spectral Decomposition of Block Skew Circulant Matrix with Skew Circulant Blocks
Let matrix be a block skew circulant matrix with skew circulant blocks as in the form of (1); then by using the properties of Kronecker products in [20], the matrix can be decomposed as where According to the style spectral decomposition of basic skew circulant (please refer to equations (10) and (11) in [12]), the style spectral decomposition of the matrix is where is an orthogonal matrix, , Consider (3) and (5); the matrix can be decomposed as Noticing that is an orthogonal matrix, so (8) is the block style spectral decomposition of the matrix .
3. The Structured Perturbation Analysis
In this section, we give the structured perturbation analysis for the block skew circulant linear systems with skew circulant blocks.
3.1. Condition Number and Relative Error of Block Skew Circulant Linear System with Skew Circulant Blocks
Consider where is defined in (1).
From (8) and the property of Kronecker products in [20], we express the matrix by using the elements in its first row as where
We denote the eigenvalues of matrix as (), and denote the eigenvalues of matrix as (); then the eigenvalues of are (regarding more properties, please refer to [20, 21])
Lemma 1. is an invertible matrix if and only if , where
Let
Theorem 2. If is a block skew circulant matrix with skew circulant blocks, then the singular values of matrix are .
Proof. Obviously, the matrix has a form as (1) and the conjugate transpose of is Through a direct calculation, we can get , and that means that is a normal matrix. By using Theorem in [22], we know that is unitarily diagonalizable. That is, there is a unitary matrix such that where (, ) are the eigenvalues of . Taking a conjugate transpose at both sides of (16), and so, we have Hence, for any , , the eigenvalues of matrix are Therefore, we can get the singular value of as Recalling (13) and (14), the proof is completed.
Since the spectral norm of matrix is defined as by using Theorem 2, we have the following corollary.
Corollary 3. Let be a block skew circulant matrix with skew circulant blocks; then the spectral norm of matrix is
By using equations (10) and (11) in [12], we can express the matrix and as where So, we can get where , and , .
Let , be the perturbation of the coefficient matrix and vector , respectively, where is a block skew circulant matrix with skew circulant blocks, has the form as follows: Let If then By using Lemma 1, we know that is an invertible matrix. Let By , , we get
Since and , so we have . Besides, we know that So, we obtain Hence, where Notice that is a block skew circulant matrix with skew circulant blocks, and . So, we get Hence, we have the following theorem.
Theorem 4. Let be defined as above. If , then where
Remark 5. From (38) and (39), the condition number of the block skew circulant linear system with skew circulant blocks can be easily computed, as well as the bound of perturbation (38).
3.2. Optimal Backward Perturbation Bound of the Block Skew Circulant Linear System with Skew Circulant Blocks
Let be an approximate solution to and let which is equivalent to Due to [14], we have
Let be an approximate solution to , where is defined in (1), as follows: Then (such as is a block skew circulant matrix with skew circulant blocks, ) Since so where , , , and , , and .
Let then which is equivalent to Hence is a convex function about , and the point of minimal value is Substituting it into (48), we can get the following.
Theorem 6. Let and be defined as above; then we have
Let be the singular value decomposition of , where and are real orthogonal matrices, , , so where , , .
Remark 7. By , we get , and hence .
Algorithm 8.
Consider the following.
Step 1. Form the block style spectral decomposition of the matrix and ,
Step 2. Compute .
Step 3. Compute .
Step 4. Compute .
Step 5. Compute .
Step 6. Form .
Step 7. Compute .
3.3. Numerical Example
In this section, we give a simple numerical example to verify the conclusions above. Suppose that , in the following example.
If the coefficient matrix of block skew circulant linear system with skew circulant blocks is , and the constant vector . Now, for comparative analysis, we give three perturbations in the following: then by equation , we get the approximate solution of correspondingly as where is the solution of and () is the solution of .
According to the Algorithm, we obtain Table 1, where means relative error of block skew circulant linear system with skew circulant blocks, is the condition number, and are obtained from the Algorithm.
From the tabular, we know that the conclusions above are right and the Algorithm is efficient.
4. Conclusion
The related problems of block skew circulant matrix with skew circulant blocks are considered in this paper. We not only present block style spectral decomposition and singular value, but also study backward perturbation analysis for the block skew circulant linear system with skew circulant blocks. The reason why we focus our attentions on block skew circulant matrix with skew circulant blocks is to explore the application of block skew circulant matrix with skew circulant blocks in the related field of medicine and real-time tracking. On the basis of existing application situation [23], we conjecture that SVD decomposition of block skew circulant matrix with skew circulant blocks will play an important role in CT-perfusion imaging of human brain. On the basis of method [7] and ideas of [24], we will exploit real-time tracking with kernel matrix of block skew circulant matrix with skew circulant blocks structure. The circulant singular value decomposition (cSVD) techniques with a block-circulant deconvolution matrix [25–29] were used to perform the deconvolution calculation to obtain the [rCBF.] curve [25, 26]. The maximum value of the [rCBF.] curve was used as the rCBF. We will exploit the skew circulant singular value decomposition (scSVD) techniques of a block skew circulant matrix with skew circulant blocks deconvolution matrix to obtain the [rCBF.] curve.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the Development Project of Science and Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11201212) and the AMEP of Linyi University, China, and the same thanks go to the works by all the authors in the references.