Applied Mathematics and Computation
The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces tr M∗M2 and tr M∗M3 are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5×5nilpotent matrix has circular numerical range are thoroughly discussed.
Matache, Valentin and Matache, Mihaela T., "When is the numerical range of a nilpotent matrix circular?" (2010). Mathematics Faculty Publications. 24.