A contribution to the normal Hankel problem

The normal Hankel problem is the one of characterizing the matrices that are normal and Hankel at the same time. This problem is far from being solved completely; only several special classes of normal Hankel matrices have been described in the literature. Recently, the authors have shown that new normal Hankel matrices could be found by seeking real solutions to systems of quadratic equations parametrized by real 2×2 matrices W with the determinant one. In this paper, we give a complete analysis of the case W=diag(α,α-1).