$A$ belongs to $n$-dimensional quaternion symmetric matrix, in the sense that $A=A^T$, where $T$ means transpose. Under transformation $U$, $A\rightarrow U\cdot A\cdot U^T$, where $U$ is $n$-dim unitary quaternion matrix, namely $U\in\mathrm{Sp}(n)$.
Then, my question is, what is the canonical form of $A$? Can we diagonalize $A$ using the above transformation such that all entries are real? What about the case that $A$ is skew-symmetric quaternion matrix? Thanks!
