  We propose for the spin density matrix two parametrizations which
automatically fulfil the non-negativity conditions, without setting any bound
on the parameters. The first one relies on a theorem, that we prove, and it is
rather simple and easily adaptable to some specific reactions, where, for
example, parity is conserved or angular momentum conservation entails selection
rules. Moreover, in the case when the rank is less than the order of the
density matrix, we show how to improve the fits to the data, either by
implementing previous suggestions, or by elaborating an alternative method, for
which we prove a second theorem. Our second parametrization is a variant of
previous treatments, it appears suitable for some particular processes.
Moreover, we discuss about the possibility of inferring the elements of the
density matrix from the differential decay width. Last, we illustrate various
examples of current interest, both in strong and weak decays; some of them may
be helpful in the investigation of physics beyond the standard model.
