Convex Noncommutative Polynomials Have Degree Two or Less

Abstract

A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided \[ p(tX+(1-t)Y) \le tp(X)+(1-t)p(Y) \] for all $0 \le t \le 1$ and for all tuples X=(X1 ,. . .,Xg and Y=(Y1 ,. . .,Yg) of symmetric matrices on a common finite dimensional vector space of a sufficiently large dimension (depending upon p). The main result of this paper is that every matrix convex polynomial has degree two or less. More generally, the polynomial p has degree at most two if convexity holds only for all matrices X and Y in an "open set." An analogous result for nonsymmetric variables is also obtained. Matrix convexity is an important consideration in engineering system theory. This motivated our work, and our results suggest that matrix convexity in conjunction with a type of "system scalability" produces surprisingly heavy constraints.