Polynomial numerical hulls of matrices

Abstract For any n -by- n complex matrix A , we use the joint numerical range W ( A , A 2 , … , A k ) to study the polynomial numerical hull of order k of A , denoted by V k ( A ) . We give an analytic description of V 2 ( A ) when A is normal. The result is then used to characterize those normal matrices A satisfying V 2 ( A ) = σ ( A ) , and to show that a unitary matrix A satisfies V 2 ( 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 , … , w n - 1 ) with w = e i 2 π / n , we determine V k ( A ) for k ∈ { 2 } ∪ { j ∈ N : j ⩾ n / 2 } . We also consider matrices A ∈ M n such that A 2 is Hermitian. For such matrices we show that V 4 ( A ) is the spectrum of A , and give a description of the set V 2 ( A ) .