  The dynamics of finite dimension open quantum systems is studied with the
help of the simplest possible form of projection operators, namely the ones
which project only onto one dimensional subspaces. The simplicity of the action
of the projection operators always leads to an analytical solution of the
dynamical master equation, even in the non-Markovian case, in any perturbative
order. The analytical solution correctly reproduces the short-time dynamics,
and can be used to recursively recover the dynamics for an arbitrary time
interval with arbitrary precision. The necessary number of relevant degrees of
freedom to completely characterise an open quantum system is $(n-1)(n+2)/2,$
where $n$ is the dimension of the Hilbert space of the open system. The method
is illustrated by two examples, the relaxation of a qubit in a thermal bath and
the dynamics of two interacting qubits in a common environment.
