The invariant straight lines of an affine transformation in ℝ^n without fixed points - Kasparek, Erwin (Annales Mathematicae Silesianae, 2010)
In this note we prove that every affine transformation in ℝn of the form f(x) = Ax+a without fixed points has an invariant straight line if and only if when g(a)∈Im(g◦g), where g(x) = (A−I)x.