  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.
