On Unitary Transposable Matrices of Order Three

A matrix $A \in M_n(\mathbb{C})$ is said to be unitarily transposable if
$$ A^T=Q^*AQ $$
for a certain unitary matrix $Q$. Every $2\times 2$ matrix is unitarily transposable; however, for greater orders, a similar statement is false, and the general description of unitarily transposable matrices of order $n$ is at present unknown. We give such a description for matrices of order three.