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, b, c, the second row will be c, a, b, and the third row will be b, c, a. This structure makes circulant matrices useful in various mathematical applications, including signal processing and solving linear equations. These matrices can be represented in a compact form using a single vector, which contains the first row's elements. The properties of circulant matrices, such as their eigenvalues and eigenvectors, are closely related to the discrete Fourier transform, making them significant in fields like linear algebra and numerical analysis.