  This paper studies the complexity of matrix Putinar's Positivstellensatz on
the semialgebraic set that is given by the polynomial matrix inequality. Under
the archimedeanness, we prove a polynomial bound on degrees of terms appearing
in the representation of matrix Putinar's Positivstellensatz. Estimates on the
exponent and constant are given. As a byproduct, a polynomial bound on the
convergence rate of matrix sum-of-squares relaxations is obtained, which
resolves an open question raised by Dinh and Pham. When the constraining set is
unbounded (the archimedeanness fails), we also prove a similar bound for the
matrix version of Putinar--Vasilescu's Positivstellensatz by exploiting
homogenization techniques.
