The normalized numerical range and the Davis–Wielandt shell

For a given n-by-n matrix A, its normalized numerical range F_N(A) is defined as the range of the function f_N,A:x↦(x^⁎Ax)/(‖Ax‖⋅‖x‖) on the complement of ker⁡A. We provide an explicit description of this set for the case when A is normal or n=2. This extension of earlier results for particular cases of 2-by-2 matrices (by Gevorgyan) and essentially Hermitian matrices of arbitrary size (by A. Stoica and one of the authors) was achieved due to the fresh point of view at F_N(A) as the image of the Davis–Wielandt shell DW(A) under a certain non-linear mapping h:R³↦C.