For certain classes of sets $S \subseteq \mathbb{R}^n$, there exist algebraic characterizations of real valued polynomials $p: \mathbb{R}^n \rightarrow \mathbb{R}$ that are positive on $S$.
Several of these algebraic characterizations, known as *positivstellensatz*, are listed here in Wikipedia.

Some of these *positivstellensatze* can be generalized for matrix polynomials, i.e. a function $P$ from $\mathbb{R}^n$ to the set of all $r \times r$ symmetric matrices whose entries are polynomials. In this case, $P$ is understood to be positive in $S$ if $P(x)$ is positive definite for all $x \in S$.

For instance, in [1], Pólya's *positivstellensatz* for homogeneous polynomials is generalized to homogeneous matrix polynomials and in [2], Putinar's *positivstellensatz* is also generalized to matrix polynomials. Stengle's and Schweighofer's *positivstellensatze* are also generalized to matrix polynomials in [3].

Does Handelman's positivstellensatz have also a matrix polynomial version?

[1]: Scherer, C. W. "Relaxations for robust linear matrix inequality problems with verifications for exactness." *SIAM Journal on Matrix Analysis and Applications 27(2)* (2005): 365-395.

[2]: Scherer, C. W. and Hol, C. W. J. "*Matrix sum-of-squares relaxations for robust semi-definite programs*." Mathematical programming 107(1-2) (2006): 189-211.

[3]: Công-Trình, Lê. "Some Positivstellensätze for polynomial matrices." *arXiv preprint* arXiv:1403.3783 (2014)