Optimal Backward Perturbation Analysis for the Block Skew Circulant Linear Systems with Skew Circulant Blocks

and Applied Analysis 3 Proof. Obviously, the matrix A has a form as (1) and the conjugate transpose of A is

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.
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,   = ((2 − 1)/), ) , ( is even) , 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 .

The Structured Perturbation Analysis
In this section, we give the structured perturbation analysis for the block skew circulant linear systems with skew circulant blocks.
where  is defined in (1).
Since the spectral norm of matrix  is defined as by using Theorem 2, we have the following corollary.
Theorem 4. Let , Â, , Δ,  min be defined as above.If Δ <  min , then where Remark 5. From ( 38) and (39), the condition number K of the block skew circulant linear system with skew circulant blocks can be easily computed, as well as the bound of perturbation (38).

Optimal Backward Perturbation Bound of the Block Skew
Circulant Linear System with Skew Circulant Blocks.Let x be an approximate solution to  =  and let which is equivalent to Due to [14], we have (‖ ‖ being any unitary invariant norm) . (42) Let x be an approximate solution to  = , where  is defined in (1), as follows: Δ is a block skew circulant matrix with skew circulant blocks} Then Ω ̸ =  (such as Δ = 0 is a block skew circulant matrix with skew circulant blocks, Δ =  x − ) Since ) where and then which is equivalent to Hence  is a convex function about ( 11 , . . .,   ), and the point of minimal value is Substituting it into (48), we can get the following.

Numerical Example.
In this section, we give a simple numerical example to verify the conclusions above.Suppose that  = 3,  = 2 in the following example.
According to the Algorithm, we obtain Table 1, where  means relative error of block skew circulant linear system with skew circulant blocks, K = max{  }/ min{  } is the condition number,  1 ( x) = ‖ −  x‖ 2 / √ 1 + ‖x‖ 2  2 and  2 are obtained from the Algorithm.
From the tabular, we know that the conclusions above are right and the Algorithm is efficient.

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 realtime 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][26][27][28][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.