Polynomial Matrix Inequality and Semidefinite Representation

Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper studies sufficient conditions guaranteeing semidefinite representability of S. We prove that S is semidefinite representable in the following cases: (i) the domain is the whole space and the matrix polynomial is matrix sos-concave; (ii) the domain is compact convex and the matrix polynomial is strictly matrix concave; (iii) the rational matrix function is q-module matrix concave on the domain. Explicit constructions of SDP representations are given. Some examples are illustrated.