  In this article we study exponential dichotomies for noninvertible linear
difference equations in finite dimensions. After giving the definition, we
study the extent to which the projection $P(k)$ in a dichotomy is unique. For
equations on $\mathbb{Z}$ it is unique but for equations on $\mathbb{Z}_+$ only
its range is unique and for $\mathbb{Z}_-$ only its nullspace.
  Here we strengthen Kalkbrenner's results and give a complete characterization
of all possible projections.
  Next we study the possibility of extending the dichotomy to a larger
interval. We reproduce the results of P\" otzsche but also show exactly when
the original projection remains unchanged.
  Next we prove that the roughness theorem, well known for additive
perturbations, holds for multiplicative perturbations also. The proof uses
ideas of Zhou, Lu and Zhang.
  Finally, following Ducrot, Magal and Seydi, we mention that the results by
Palmer on finite time conditions on dichotomy for the invertible case can be
extended to the noninvertible case.
