  We study the semiclassical time evolution of observables given by matrix
valued pseudodifferential operators and construct a decomposition of the
Hilbert space $L^2(\rz^d)\otimes\kz^n$ into a finite number of almost invariant
subspaces. For a certain class of observables, that is preserved by the time
evolution, we prove an Egorov theorem. We then associate with each almost
invariant subspace of $L^2(\rz^d)\otimes\kz^n$ a classical system on a product
phase space $\TRd\times\cO$, where $\cO$ is a compact symplectic manifold on
which the classical counterpart of the matrix degrees of freedom is
represented. For the projections of eigenvectors of the quantum Hamiltonian to
the almost invariant subspaces we finally prove quantum ergodicity to hold, if
the associated classical systems are ergodic.
