Differential properties of the numerical range map of pairs of matrices

Let A = (A1, A2) be a pair of Hermitian operators in n and A = A1 + iA2. We investigate certain differential properties of the numerical range map nA : x → ( A1x, x , A2x, x ) with the aim of better understanding the nature of the numerical range W(A) of A. For example, the joint eigenvalues of A correspond to the stationary points of nA (i.e. points where the derivative n′A vanishes). Moreover, points x where rank n′A(x) = 2 get mapped by nA into the interior W(A)° of W(A). For n = 2, it turns out that if A1 and A2 have no common invariant subspace, then the image under nA of the set Σ1(A) consisting of those points x with rank n′A(x) = 1 is precisely the boundary ∂W(A) of W(A), and the image under nA of the rank 2 points for n′A is precisely W(A)°; there are no rank 0 points for n′A. As a consequence (for n = 2) we have that A1A2 = A2A1 iff Σ1(A) ≠ nA−1(∂W(A)).