The upper numerical range of a quaternionic matrix is not a complex numerical range Original Research Article

A recent result is that the quaternionic numerical range of a matrix with quaternion entries has a convex intersection with the upper half complex plane. This intersection is now shown to be generally not achievable as the upper half plane part of the complex numerical range of any complex matrix. A key step in the proof is that if a complex matrix has an elliptical arc as part of the boundary of its complex numerical range, then the full ellipse defined by the arc is also in its complex numerical range.