  We show under general conditions that the linearized force-based
quasicontinuum (QCF) operator has a positive spectrum, which is identical to
the spectrum of the quasinonlocal quasicontinuum (QNL) operator in the case of
second-neighbour interactions. Moreover, we establish a bound on the condition
number of a matrix of eigenvectors that is uniform in the number of atoms and
the size of the atomistic region. These results establish the validity of and
improve upon recent conjectures ([arXiv:0907.3861, Conjecture 2] and
[arXiv:0910.2013, Conjecture 8]) which were based on numerical experiments.
  As immediate consequences of our results we obtain rigorous estimates for
convergence rates of (preconditioned) GMRES algorithms, as well as a new
stability estimate for the QCF method.
