A linear autonomous system of difference equations x_k+1 = Ax_k can be transformed to its Jordan normal form, i.e., the transformed system is in block diagonal form and the blocks correspond to different eigenvalues. We generalize this result to arbitrary nonautonomous linear systems x_k+1 = A_kx_k with invertible matrices A_k ∈ ℝ^{N × N}, k ∈ {..., -1, 0, 1,...}.