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Let A\,... ,Am benxn hermitian matrices and let Un be the real space ofnXn hermitian matrices. If span{ A\,..., Am } = Wn, then the ex- treme rays of the joint angular field of values of {A\,..., Am} are determined. Then this cone is used to give necessary and sufficient conditions for the exis- tence of hermitian matrices B\,..., Bm such that Ai ® Bi + • ■ • + Am (g> Bm preserves the cone of the positive semidefinite matrices where A ® B is the dyad product A B(H) = (trBH)A. 1. Introduction. Let V be a finite-dimensional real vector space of dimension n with an inner product ( , ). A subset K C V is called a (convex) cone iff for all a, s > 0 and x,y e K we have ax + sy e K. We call K closed if it is a closed set in the natural topology of V. The cone K is pointed iff K 0 (—K) = {0} (i.e., K contains no nonzero subspace) and full iff K has nonempty interior. In finite- dimensional spaces it is well known that K is full iff K — K = V. A face F of a cone K is a subcone such that x, y — x G K and y G F implies x G F. The dimension of a face F, dimF, is the dimension of the linear span of F. If dimi1 = 1 we call F an extreme ray of K. If S C K, then (x), and if 0, Vx G K}. Next if Hom(V) denotes the vector space of linear mappings of V, then define

Applications of the joint angular field of values