  This paper studies the matrix Moment-SOS hierarchy for solving polynomial
matrix optimization. Our first result is to show the tightness (i.e., the
finite convergence) of this hierarchy, if the nondegeneracy condition, strict
complementarity condition and second order sufficient condition hold at every
minimizer, under the usual archimedeanness assumption. A useful criterion for
detecting tightness is the flat truncation. Our second result is to show that
every minimizer of the moment relaxation must have a flat truncation when the
relaxation order is big enough, under the above mentioned optimality
assumptions. These results give connections between nonlinear semidefinite
optimization theory and Moment-SOS methods for solving polynomial matrix
optimization.
