We discuss the integrable hierarchies that appear in multi-matrix mode
ls. They can be envisaged as multi-field representations of the KP hie
rarchy. We then study the possible reductions of this systems via the
Dirac reduction method by suppressing successively one by one part of
the fields. We find in this way new integrable hierarchies, of which w
e are able to write the Lax pair representations by means of suitable
Drinfeld-Sokolov linear systems. At the bottom of each reduction proce
dure we find an Nth KdV hierarchy. We discuss in detail the case which
leads to the KdV hierarchy and to the Boussinesque hierarchy, as well
as the general case in the dispersionless limit.