We first give the style spectral decomposition of a special skew circulant matrix and then get the style decomposition of arbitrary skew circulant matrix by making use of the Kronecker products between the elements of first row in skew circulant and the special skew circulant . Besides that, we obtain the singular value of skew circulant matrix as well. Finally, we deal with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix on the base of its style spectral decomposition. 1. Introduction A skew circulant matrix with the first row is a square matrix of the form denoted by . Skew circulant matrices have important applications in various disciplines including image processing, signal processing, solving Toeplitz matrix problems, and preconditioner. The skew circulant matrices are considered as preconditioners for linear-multistep-formulae (LMF-) based ordinary differential equations (ODEs) codes; Hermitian and skew-Hermitian Toeplitz systems are considered in [1–4]. Lyness and S？revik [5] employed a skew circulant matrix to construct -dimensional lattice rules. Spectral decompositions of skew circulant and skew left circulant matrices are discussed in [6]. Akhondi and Toutounian [7] presented a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. Narasimha [8] believed that the linear convolution required in block filtering can be decomposed into a sum of skew-circulant convolutions and such convolutions can be realized efficiently with half-length complex transforms when the signals are real. Liu and Vaidyanathan [9] presented a new family of normal form state-space structures, the method used allows people to synthesize in normal form, most IIR transfer functions, and the state transition matrices involved are either circulant or skew circulant matrices. Vaidyanathan and Pal [10] examined a case where two arrays are generated by matrices that are adjugates of each other; in this case, it is possible to obtain a dense rectangular tiling of the 2 frequency plane from a pair of coarse 2 DFT filter banks; the special case where the adjugate pairs are generated by skew circulant matrices has some advantages, which are examined in detail. An additional convolution-multiplication property for the skew-circulant convolution operation , where is a skew-circulant matrix; besides, skew-circulant convolution is the underlying form of convolution in half of the 40 cases of symmetric convolution, and the convolution is an extension of a result Vernet's
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