On the convexity and circularity of the numerical range of nilpotent quaternionic matrices - Carvalho, L.; Mendes, S.; Diogo, C. (New York Journal of Mathematics, 2019)
We provide a sufficient condition for the numerical range of a nilpotent matrix N to be circular in terms of the existence of cycles in an undirected graph associated with N. We prove that if we add to this matrix N a diagonal real matrix D, the matrix D+N has convex numerical range. For 3 x 3 nilpotent matrices, we strength further our results and obtain necessary and sufficient conditions for circularity and convexity of the numerical range.