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In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform.〔Davis, Philip J., Circulant Matrices, Wiley, New York, 1970 ISBN 0471057711〕 They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group and hence frequently appear in formal descriptions of spatially invariant linear operations. In cryptography, a circulant matrix is used in the MixColumns step of the Advanced Encryption Standard. ==Definition== An circulant matrix takes the form : A circulant matrix is fully specified by one vector, , which appears as the first column of . The remaining columns of are each cyclic permutations of the vector with offset equal to the column index. The last row of is the vector in reverse order, and the remaining rows are each cyclic permutations of the last row. Note that different sources define the circulant matrix in different ways, for example with the coefficients corresponding to the first row rather than the first column of the matrix, or with a different direction of shift. The polynomial is called the ''associated polynomial'' of matrix . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「circulant matrix」の詳細全文を読む スポンサード リンク
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