  We investigate a decomposition of a unital Lindblad dynamical map of an open
quantum system into two distinct types of mapping on the Hilbert-Schmidt space
of quantum states. One component of the decomposed map corresponds to
reversible behaviours, while the other to irreversible characteristics. For a
finite dimensional system, we employ real vectors or Bloch representations and
express a dynamical map on the state space as a real matrix acting on the
representation. It is found that rotation and scaling transformations on the
real vector space, obtained from the real-polar decomposition, form building
blocks for the dynamical map. Consequently, the change of the linear entropy or
purity, which indicates dissipative behaviours, depends only on the scaling
part of the dynamical matrix. The rate of change of the entropy depends on the
structure of the scaling part of the dynamical matrix, such as eigensubspace
partitioning, and its relationship with the initial state. In particular, the
linear entropy is expressed as a weighted sum of the exponential-decay
functions in each scaling component, where the weight is equal to
$\vert\vec{x}_k(\rho)\vert^2$ of the initial state $\rho$ in the subspace. The
dissipative behaviours and the partition of eigensubspaces in the decomposition
are discussed and illustrated for qubit systems.
