  We study the invariants of arbitrary dimensional multipartite quantum states
under local unitary transformations. For multipartite pure states, we give a
set of invariants in terms of singular values of coefficient matrices. For
multipartite mixed states, we propose a set of invariants in terms of the trace
of coefficient matrices. For full ranked mixed states with nondegenerate
eigenvalues, this set of invariants is also the necessary and sufficient
conditions for the local unitary equivalence of such two states.
