Unitary equivalence to a complex symmetric matrix: low dimensions

A matrix \(T \in \M_n(\C)\) is \emph{UECSM} if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize \(4 \times 4\) nilpotent matrices which are UECSM and we settle an open problem which has lingered in the \(3 \times 3\) case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above