Polynomial numerical hulls of matrices

For any n-by-n complex matrix A, we use the joint numerical range W(A,A2,…,Ak) to study the polynomial numerical hull of order k of A, denoted by Vk(A). We give an analytic description of V2(A) when A is normal. The result is then used to characterize those normal matrices A satisfying V2(A)=σ(A), and to show that a unitary matrix A satisfies V2(A)=σ(A) if and only if its eigenvalues lie in a semicircle, where σ(A) denotes the spectrum of A. When A=diag(1,w,…,wn-1) with w=ei2π/n, we determine Vk(A) for k∈{2}∪{j∈N:j⩾n/2}. We also consider matrices A∈Mn such that A2 is Hermitian. For such matrices we show that V4(A) is the spectrum of A, and give a description of the set V2(A).