Line segments and elliptic arcs on the boundary of a numerical range
Hwa-Long Gau, Pei Yuan Wu · Linear and Multilinear Algebra · 2007
For an n-by-n complex matrix A, we consider the numbers of line segments and elliptic arcs on the boundary ∂W(A) of its numerical range. We show that (a) if and A has an (n − 1)-by-(n − 1) submatrix B with W(B) an elliptic disc, then there can be at most 2n − 2 line segments on ∂W(A), and (b) if , then ∂W(A) contains at most (n − 2) arcs of any ellipse. Moreover, both upper bounds are sharp. For nilpotent matrices, we also obtain analogous results with sharper bounds.