On the generalized numerical range

Let Ak , k − 1, …, m be n × n Hermitian matrices and let have components fk (x) = xHAkx, k=1,…,m. When n ⩾ 3 and m = 3, the set W(A1 … Am )= |f(x) x 2 − 1 | is convex. This property does not hold in general when m > 3. These particular cases of known results are proven here using a direct, geometric approach. A geometric characterizarion of the contact surfaces is obtained for any n and m. Necessary conditions are given for f(x) to be on the boundary of W(A, …Am ) or on certain subsets of this boundary. These results are of interest in the context of the computation of the structured singular value, a recently introduced tool for the analysis and synthesis of control systems.