Block diagonal dominance and reducibility for linear differential systems

We consider the linear homogeneous differential equation x′(t) = A(t)x(t). Conditions are given on the entries of A(t) under which the equation is reducible. It is shown that if A(t) is almost periodic the reduction is via an almost periodic change of variables. The result on reducibility is used to give new conditions on A(t) under which the equation admits an exponential dichotomy and to prove a result on the reducibility of almost periodic perturbations of constant coefficient systems.