Every Convex Free Basic Semi-algebraic Set Has an Lmi Representation

The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex non-commutative basic open semi-algebraic set (defined below). The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and systems engineering. A non-commutative basic open semi-algebraic set is defined in terms of a non-commutative `×`-matrix polynomial p(x1 · · · , xg). Such a polynomial is a linear combinations of words in non-commuting free variables {x1, . . . , xg} with coefficients from M`, the ` × ` matrices (for some `). The involution T on words given by sending a concatenation of letters to the same letters, but in the reverse order (for instance (xjx`) T = x`xj), extends naturally to such polynomials and p is itself symmetric if p = p. Let Sn(R) denote the set of g-tuples X = (X1, . . . , Xg) of symmetric n × n matrices. A polynomial can naturally be evaluated on a tuple X ∈ Sn(R) yielding a value p(X) which is an ` × ` block matrix with n×n matrix entries. Evaluation at X is compatible with the involution since p (X) = p(X) and if p is symmetric, then p(X) is a symmetric matrix. Assuming that p(0) is invertible, the invertibility set Dp(n) of a noncommutative symmetric polynomial p in dimension n is the component of 0 of the set {X ∈ Sn(R) : p(X) is invertible}. The invertibility set, Dp, is the sequence of sets (Dp(n)), which is the type of set we call a nc basic open semi-algebraic set. The noncommutative set Dp is called convex if, for each n, Dp(n) is convex. A linear matrix inequality is the special case where p = L is an affine linear symmetric polynomial with L(0) = I. In this case, DL is clearly convex. A set is said to have a Linear Matrix Inequality Representation if it is the set of all solutions to some LMI, that is, it has the form DL for some L. The main theorem says: if p(0) is invertible and Dp is bounded, then Dp has an LMI representation if and only if Dp is convex. Date: January 5, 2010. 1991 Mathematics Subject Classification. 47Axx (Primary). 47A63, 47L07, 47L30, 14P10 (Secondary).