On the Generalized Numerical Range.

Abstract

Let A_k, k = 1, ...., m be n x n Hermitian matricies and let f: C^n --> R^m have components f^k(x) = x^H A_k(x), k = 1, ..., m. When n >= 3 and m = 3, the set W(A_1>,..., A_m) = {f(x): || x || = 1} PROPER subset of R_m is convex. This property does not hold in general when m > 3. These particuar cases of known results are proven here using a direct, geometric approach. A geometric characterization of the contact surfaces is obtained for any n and m. Necessary conditions are given for f(x) to be on boundary of W(A_1,..., A_2) or on the certain subsets of this boundary. These results are of interest in the context of the computations of the structured singular value, a recently introduced tool for the analysis and synthesis of control systems.