  The generic linear evolution of the density matrix of a system with a
finite-dimensional state space is by stochastic maps which take a density
matrix linearly into the set of density matrices. These dynamical stochastic
maps form a linear convex set that may be viewed as supermatrices. The property
of hermiticity of density matrices renders an associated supermatrix hermitian
and hence diagonalizable; but the positivity of the density matrix does not
make this associated supermatrix positive. If it is positive, the map is called
completely positive and they have a simple parametrization. This is extended to
all positive (not completely positive) maps. A contraction of a norm-preserving
map of the combined system can be contracted to obtain all dynamical maps. The
reconstruction of the extended dynamics is given.
