Numerical range circumscribed by two polygons Original Research Article

We show that, for any 2n+2 distinct points a1,a1′,a2,a2′,…,an+1,an+1′ (in this order) on the unit circle, there is an n-by-n matrix A, unique up to unitary equivalence, which has norm one and satisfies the conditions that it has all its eigenvalues in the open unit disc, In−A*A has rank one and its numerical range is circumscribed by the two (n+1)-gons a1a2cdots, three dots, centeredan+1 and a1′a2′cdots, three dots, centeredan+1′. This generalizes the classical result of the existence of a conical curve circumscribed by two triangles which are already inscribed on another conical curve.