Linear images of joint unitary orbits of Hermitian matrices

In this paper, interplay between the geometric properties of WL(A) and algebraic properties of linear maps L:Hnm→Rℓ and A∈Hnm will be studied. We shall give a characterization of L such that rL(⋅) defines a norm on Hnm. Furthermore, we shall also characterize linear maps L:Hnm→Rℓ and A∈Hnm in which WL(A) is a singleton set. More generally, the characterization of linear maps L:Hnm→Rℓ and A∈Hnm in which WL(A) lies inside a k-dimensional affine subspace of Rℓ will be given.