The normalized numerical range and the Davis-Wielandt shell

For a given n-by-n matrix A, its normalized numerical rangeFN(A) is defined as the range of the function fN,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 FN(A) as the image of the Davis-Wielandt shell DW(A) under a certain non-linear mapping h:R3↦C.