is an eigenvector.) The corresponding eigenvalues are given by Determinant As a consequence of the explicit formula for the eigenvalues above, the determinant of a circulant matrix can be computed as: Since taking the transpose does not change the eigenvalues of a matrix, an equivalent formulation is Rank The rank of a circulant matrix is equal to , where is the degree of the polynomial . Other properties Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix : where is given by The set of circulant matrices forms an -dimensional vector space with respect to addition and scalar multiplication. This space can be interpreted as the space of functions on the cyclic group of order , , or equivalently as the group ring of . Circulant matrices form a commutative algebra, since for any two given circulant matrices and , the sum is circulant, the product is circulant, and . For a nonsingular circulant matrix , its inverse is also circulant. For a singular circulant matrix, its Moore–Penrose pseudoinverse is circulant. The matrix that is composed of the eigenvectors of a circulant matrix is related to the discrete Fourier transform and its inverse transform: Consequently the matrix diagonalizes . In fact, we have where is the first column of . The eigenvalues of are given by the product . This product can be readily calculated by a fast Fourier transform. Conversely, for any diagonal matrix , the product is circulant. Let be the (monic) characteristic polynomial of an circulant matrix , and let be the derivative of . Then the polynomial is the characteristic polynomial of the following submatrix of : (see for the proof). Analytic interpretation Circulant matrices can be interpreted geometrically, which explains the connection with the discrete Fourier transform. Consider vectors in as functions on the integers with period , (i.e., as periodic bi-infinite sequences: ) or equivalently, as functions on the cyclic group of order ( or ) geometrically, on (the vertices of) the regular : this is a discrete analog to periodic functions on the real line or circle. Then, from the perspective of operator theory, a circulant matrix is the kernel of a discrete integral transform, namely the convolution operator for the function ; this is a discrete circular convolution. The formula for the convolution of the functions is (recall that the sequences are periodic) which is the