  Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on
$m$-columns of half-densities on a closed manifold $M$, whose principal symbol
is assumed to have simple eigenvalues. We show existence and uniqueness of $m$
orthonormal pseudodifferential projections commuting with the operator $A$ and
provide an algorithm for the computation of their full symbols, as well as
explicit closed formulae for their subprincipal symbols. Pseudodifferential
projections yield a decomposition of $L^2(M)$ into invariant subspaces under
the action of $A$ modulo $C^\infty(M)$. Furthermore, they allow us to decompose
$A$ into $m$ distinct sign definite pseudodifferential operators. Finally, we
represent the modulus and the Heaviside function of the operator $A$ in terms
of pseudodifferential projections and discuss physically meaningful examples.
