Unitary equivalence to a complex symmetric matrix: Low dimensions

Abstract

A matrix $T\in {\mathbf{M}}_n(\mathbb{C})$ is 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.