  In this paper, we characterize completely the structure of Clifford
semigroups of matrices over an arbitrary field. It is shown that a semigroups
of matrices of finite order is a Clifford semigroup if and only if it is
isomorphic to a subdirect product of some linear (0-)groups. Then we generalize
this result to a semigroup of matrices of countably infinite order and give a
similar necessary and sufficient condition for it to be a Clifford semigroup.
