Unitary equivalence of a matrix to its transpose

TL;DR

This paper investigates the conditions under which complex matrices are unitarily equivalent to their transpose, revealing size-dependent differences in their canonical forms and challenging previous assumptions for larger matrices.

Contribution

It provides a canonical decomposition for matrices unitarily equivalent to their transpose and shows the failure of a naive equivalence assertion for matrices of size 8x8 and larger.

Findings

The canonical decomposition for UET matrices is established.

The naive equivalence to complex symmetric matrices holds only for matrices 7x7 or smaller.

The equivalence fails for matrices 8x8 and larger.

Abstract

Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naive assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix (i.e., $T = T^t$) holds for matrices 7x7 and smaller, but fails for matrices 8x8 and larger.