  In this paper, we derive a new linear convergence rate for the gradient
method with fixed step lengths for non-convex smooth optimization problems
satisfying the Polyak-Lojasiewicz (PL) inequality. We establish that the PL
inequality is a necessary and sufficient condition for linear convergence to
the optimal value for this class of problems. We list some related classes of
functions for which the gradient method may enjoy linear convergence rate.
Moreover, we investigate their relationship with the PL inequality.
