  The (matricial) 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, e.g., H-infinity 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.
