Affine permutations can be represented as infinite periodic permutation matrices. If u : Z → → Z is an affine permutation, the corresponding matrix has entry 1 at position ( i , u ( i ) ) in the infinite grid Z × × Z for each integer i, and all other entries are equal to 0. Since u is a bijection, the resulting matrix contains exactly one 1 in every row and column. The periodicity condition on the map u ensures that the entry at position ( a , b ) is equal to the entry at position ( a + n , b + n ) for every pair of integers ( a , b ) . For example, a portion of the matrix for the affine permutation [ 2 , 0 , 4 ] ∈ ∈ S ~ ~ 3 is shown in the figure. In row 1, there is a 1 in column 2; in row 2, there is a 1 in column 0; and in row 3, there is a 1 in column 4. The rest of the entries in those rows and columns are all 0, and all the other entries in the matrix are fixed by the periodicity condition.