An n×n matrix whose rows are composed of cyclically shifted versions of a length-n list l. For example, the 4×4 circulant matrix on the list l = {1, 2, 3, 4} is given by C = [4 | 1 | 2 | 3 3 | 4 | 1 | 2 2 | 3 | 4 | 1 1 | 2 | 3 | 4]. Circulant matrices are very useful in digital image processing, and the n×n circulant matrix is implemented as CirculantMatrix[l, n] in the Mathematica application package Digital Image Processing. Circulant matrices can be implemented in the Wolfram Language as follows.

circulant determinant