Unitary equivalence to a complex symmetric matrix: low dimensions

TL;DR

This paper investigates when matrices in low dimensions are unitarily equivalent to complex symmetric matrices, providing complete characterizations for certain cases and highlighting fundamental differences between three and higher dimensions.

Contribution

It develops techniques for low-dimensional cases, fully characterizes 4x4 nilpotent matrices as UECSM, and resolves an open problem in 3x3 matrices.

Findings

Complete characterization of 4x4 nilpotent UECSM matrices

Resolution of an open problem in 3x3 case

Identification of key differences between 3D and higher dimensions

Abstract

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