  It is shown that each linear operator on a separable Hilbert space which
generates a finite type I von Neumann algebra has, up to unitary equivalence, a
unique representation as a direct integral of inflations of mutually unitary
inequivalent irreducible matrices. This leads to a simplification of the
so-called prime (or central) decomposition and the multiplicity theory for such
operators. The concept of so-called p-isomorphisms between special classes of
such operators is discussed. All results are formulated in more general
settings; that is, for tuples of closed densely defined operators affiliated
with finite type I von Neumann algebras.
