On the convexity and circularity of the numerical range of nilpotent quaternionic matrices (Q2280164)

For a \(n\times n\) matrix \(A\) with quaternionic entries, its numerical range is the set of Hamilton quaternions of the form \(x^\ast Ax\), where \(x\) runs over the unit sphere. The graph associated to the matrix \(A\) has \(n\) vertices with two vertices \(i\) and \(j\) adjacent whenever either \(A_{ij}\neq 0\) or \(A_{ji}\neq 0\). The authors first prove that if the graph associated to a nilpotent matrix \(A\) is a tree, then the numerical range of \(A\) is a disk, and further that if \(A\) is unitary equivalent to a sum of a real diagonal matrix and a nilpotent upper triangular cycle-free matrix, then its numerical range is a union of disks. In the case of \(3\times 3\) matrices, these results are further brought to the characterization which states that the numerical range of a \(3\times 3\) quaternionic matrix \(A\) is a disk if and only if the associated graph of \(A\) is cycle-free.