Non-commutative polynomials with convex level slices

Abstract

Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The polynomial p is symmetric if it takes symmetric values. Under natural irreducibility assumptions and other mild hypothesis, the article gives an algebraic certificate for symmetric polynomials p with the property that for sufficiently many tuples A, the set of those tuples X such that p(A,X) is positive definite is convex. In particular, p has degree at most two in x. The case of noncommutative quasi-convex polynomials is of particular interest. The problem analysed here occurs in linear system engineering problems. There the A tuple corresponds to the parameters describing a system one wishes to control while the X tuple corresponds to the parameters one seeks in designing the controller. In this setting convexity is typically desired for numerical reasons and to guarantee that local optima are in fact global. Further motivation comes from the theories of matrix convexity and operator systems