In linear algebra, a circulant matrix is a square matrix in which all row vectors are composed of the same elements and each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplitz matrix. 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. 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. This property is also critical in modern software defined radios, which utilize Orthogonal Frequency Division Multiplexing to spread the symbols (bits) using a cyclic prefix. This enables the channel to be represented by a circulant matrix, simplifying channel equalization in the frequency domain. In cryptography, a circulant matrix is used in the MixColumns step of the Advanced Encryption Standard. Definition An circulant matrix takes the form or the transpose of this form (by choice of notation). When the term is a square matrix, then the matrix is called a block-circulant matrix. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of . The remaining columns (and rows, resp.) of are each cyclic permutations of the vector with offset equal to the column (or row, resp.) index, if lines are indexed from 0 to . (Cyclic permutation of rows has the same effect as cyclic permutation of columns.) The last row of is the vector shifted by one in reverse. Different sources define the circulant matrix in different ways, for example as above, or with the vector corresponding to the first row rather than the first column of the matrix; and possibly with a different direction of shift (which is sometimes called an anti-circulant matrix). The polynomial is called the associated polynomial of matrix . Properties Eigenvectors and eigenvalues The normalized eigenvectors of a circulant matrix are the Fourier modes, namely, where is a primitive -th root of unity and is the imaginary unit. (This can be understood by realizing that multiplication with a circulant matrix implements a convolution. In Fourier space, convolutions become multiplication. Hence the product of a circulant matrix with a Fourier mode yields a multiple of that Fourier mode, i.e. it