  Given the projections of two semialgebraic sets defined by polynomial matrix
inequalities, it is in general difficult to determine whether one is contained
in the other. To address this issue we propose a new matrix Positivstellensatz
that uses lifting polynomials. Under the classical archimedean condition and
some mild natural assumptions, we prove that such a containment holds if and
only if the proposed matrix Positivstellensatz is satisfied. The corresponding
certificate can be searched for by solving a semidefinite program. An important
application is to certify when a spectrahedrop (i.e., the projection of a
spectrahedron) is contained in another one.
