Polynomial numerical hulls of the direct sum of a normal matrix and a Jordan block

Let Jk(λ) be the k×k Jordan block with eigenvalue λ and let N be an m×m normal matrix. In this paper we study the polynomial numerical hulls of order 2 and n−1 for A=Jk(λ)⊕N, where n=m+k. We obtain a necessary and sufficient condition such that V2(A) has an interior point. Also, we analytically characterize V2(J2(λ)⊕N) and we show that if σ(N)∪{λ} is co-linear, then V2(J2(λ)⊕N)=⋃a∈σ(N)V2(J2(λ)⊕[a]). Finally, we study Vn−1(A) and we show that if σ(N) is neither co-linear nor co-circular, then Vn−1(A) has at most one point more than σ(A).