Ky Fan's Dominance Theorem and Eaton Triple

A generalization of the Dominance Theorem of Ky Fan on the unitarily invariant norm is obtained in the content of Eaton triple. The results of Zietak on the characterization of the set of dual matrices of a given matrix and the faces of the unit ball, both with respect to a unitarily invariant norm, are also extended in the same context. The notion of dual matrices of a given matrix A with respect to a unitarily invariant norm is related to the subdifferential of the norm at A. A generalization of So's characterization of the extreme points of the unit ball is given.