Structure of nilpotent matrices over fields

A zero-nonzero pattern A is said to be potentially nilpotent over a field F if there exists a nilpotent matrix with entries in F having zero-nonzero pattern A. We explore the construction of potentially nilpotent patterns over a field. We present classes of patterns which are potentially nilpotent over a field F if and only if the field F contains certain roots of unity. We then introduce some sparse patterns of order n ≥ 4 which are spectrally arbitrary over C but not over R. We also identify all irreducible patterns of order four which are potentially nilpotent over R or C.