Suppose we have some random matrix $A \in M_{n\times n}(\mathbb{C})$. Now I wish to claim there exists a unitary matrix $U \in M_{n\times n}(\mathbb{C})$ such that $UA$ is self-adjoint, i.e. $(UA)^{*} = UA$.
This yields $UA = (UA)^{*} = A^{*}U^{*}$, but I don't know it that is of any use. Writing it out for $n=2$ seems to work, but I was hoping there is a theorem (which I have been unable to find) which states the existence for general $n$?