A circulant matrix is fully specified by one vector, formula_5, which appears as the first column (or row) of formula_3. The remaining columns (and rows, resp.) of formula_3 are each cyclic permutations of the vector formula_5 with offset equal to the column (or row, resp.) index, if lines are indexed from 0 to formula_9. (Cyclic permutation of rows has the same effect as cyclic permutation of columns.) The last row of formula_3 is the vector formula_5 shifted by one in reverse.