On the complexity of matrix Putinar's Positivstellensatz

This paper studies the complexity of matrix Putinar's Positivstellens{\"a}tz on the semialgebraic set that is given by the polynomial matrix inequality. \rev{When the quadratic module generated by the constrained polynomial matrix is Archimedean}, we prove a polynomial bound on the degrees of terms appearing in the representation of matrix Putinar's Positivstellens{\"a}tz. 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, we also prove a similar bound for the matrix version of Putinar--Vasilescu's Positivstellens{\"a}tz by exploiting homogenization techniques.