Unitary equivalence of a matrix to its transpose (Q2919622)

The authors discuss a problem introduced by \textit{P. R. Halmos} [Linear algebra problem book. Washington, DC: MAA, Math. Association of America (1995; Zbl 0846.15001); Proposition 159] and prove a remarkable result concerning this problem. They prove that a matrix \(T\in M_n(\mathbb C)\) is unitarily equivalent to its transpose \(T^t\) if and only if it is unitarily equivalent to a direct sum of (some of the summands may be absent):{\parindent=1.0cm\begin{itemize}\item[(i)] irreducible complex symmetric matrices;\item[(ii)] irreducible skew-Hamiltonian matrices (such matrices are necessarily \(8\times 8\) or larger);\item[(iii)] \(2d\times 2d\) blocks of the form \(\left(\begin{matrix} A &\;0\\ 0 &\;A^t\end{matrix}\right)\), where \(A\in M_d(\mathbb C)\) is irreducible and neither unitarily equivalent to a complex symmetric matrix nor unitarily equivalent to a skew-Hamiltonian matrix (such matrices are necessarily \(6\times 6\) or larger).NEWLINENEWLINE\end{itemize}} Moreover, the unitary orbits of the three classes described above are pairwise disjoint.