Affine iteration ($x_{k+1}=M{x}_k+c$)

Iterating an affine map gives $x_k=M^{k-1}{x}_1+{\sum }_{j=0}^{k-2}{M}^{j}c$. A fixed point exists exactly when $c$ is in the image of a linear map $I-M$. If the spectral radius of $M$ is below one, its matrix powers tend to zero and every orbit converges to the unique fixed point $(I-M{)}^{-1}c$.