Polynomial numerical hulls of matrices

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) = sigma (A), and to show that a unitary matrix A satisfies V-2 (A) = sigma (A) if and only if its eigenvalues lie in a semicircle, where sigma (A) denotes the spectrum of A. When A = diag(1, w,..., w(n-1)) with w = e(i2 pi/n), we determine V-k(A) for k is an element of {2} boolean OR {j is an element of N: j >= n/2}. We also consider matrices A is an element of 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). (C) 2007 Published by Elsevier Inc.