The (matrical) solution set of a Linear Matrix Inequality (LMI) is a convex
basic non-commutative semi-algebraic set. The main theorem of this paper is a
converse, a result which has implications for both semidefinite programming and
systems engineering. For p(x) a non-commutative polynomial in free variables x=
(x1, ... xg) we can substitute a tuple of symmetric matrices X= (X1, ... Xg)
for x and obtain a matrix p(X). Assume p is symmetric with p(0) invertible, let
Ip denote the set {X: p(X) is an invertible matrix}, and let Dp denote the
component of Ip containing 0.

THEOREM: If the set Dp is uniformly bounded independent of the size of the
matrix tuples, then Dp has an LMI representation if and only if it is convex.
Linear engineering systems problems are called "dimension free" if they can be
stated purely in terms of a signal flow diagram with L2 performance measures,
eg. Hinfinity control. Conjecture: A dimension free problem can be made convex
if and only it can be made into an LMI. The theorem here settles the core case
affirmatively.