Numerical range of a normal compression II

As in the predecessor [Numerical range of a normal compression, Linear and Multilinear Algebra, in press] of this paper, we consider properties of matrices of the form V*NV, where N=diag(a₁,⋯,a _n+1) is a diagonal matrix with distinct eigenvalues a_js such that each of them is a corner of the convex hull they generate, and V is an (n+1)-by-n matrix with V*V=I_n such that any nonzero vector orthogonal to the range space of V has all its components nonzero. We obtain that such a matrix A is determined by its eigenvalues up to unitary equivalence, is irreducible and cyclic, and the boundary of its numerical range is a differentiable curve which contains no line segment. We also consider the condition for the existence of another matrix of the above type which dilates to A such that their numerical ranges share some common points with the boundary of the (n+1)-gon a₁⋯a_n+1.