A circulant matrix is a special type of square matrix where each row is a cyclic shift of the row above it. For example, if the first row is [a_0, a_1, a_2], the second row will be [a_2, a_0, a_1], and the third row will be [a_1, a_2, a_0]. This structure makes circulant matrices useful in various mathematical applications, including signal processing and solving linear equations. Circulant matrices have unique properties, such as being diagonalizable by the Discrete Fourier Transform (DFT). Their eigenvalues can be easily computed, which simplifies many calculations in linear algebra. Additionally, they are closely related to Toeplitz matrices, which have constant values along diagonals, but circulant matrices specifically maintain a cyclic pattern across rows.