  We give a complete characterization of the behavior of the Anderson
acceleration (with arbitrary nonzero mixing parameters) on linear problems. Let
n be the grade of the residual at the starting point with respect to the matrix
defining the linear problem. We show that if Anderson acceleration does not
stagnate (that is, produces different iterates) up to n, then the sequence of
its iterates converges to the exact solution of the linear problem. Otherwise,
the Anderson acceleration converges to the wrong solution. Anderson
acceleration and of GMRES are essentially equivalent up to the index where the
iterates of Anderson acceleration begin to stagnate. This result holds also for
an optimized version of Anderson acceleration, where at each step the mixing
parameter is chosen so that it minimizes the residual of the current iterate.
