TITLE: Unitary matrix as a product of real orthogonal and complex symmetric
QUESTION [5 upvotes]: Show that any unitary matrix $U$ can be written as a product of real orthogonal matrix and complex symmetric matrix.
Hint: For any unitary matrix $A$ and for any $n\in \mathbb{N}$ there is unitary matrix $B$ such that $B^n=A$. (I proved it here)
My attempt: Since $U$ is unitary then $U^TU$ is also unitary matrix then by hint one can find unitary matrix $X$ such that $X^2=U^TU$. Then I stucked and don't know how to proceed.
Would be thankful if someone can show how to solve this problem.

REPLY [3 votes]: Let $X$ be a unitary matrix with $X^2 = UU^T$, and $X = p(UU^T)$ (for some polynomial $p$).  Note that $X$ must itself be symmetric. Because $X$ commutes with $UU^T$, $X^{-1}$ also commutes with $UU^T$.
We have $U = X \cdot (X^{-1}U)$.  We can see that $X^{-1}U$ is orthogonal as follows:
$$
(X^{-1}U)(X^{-1}U)^T = X^{-1} UU^T X^{-T} = X^{-1}(UU^T) X^{-1} 
\\= (UU^T)X^{-1}X^{-1} = UU^T (UU^T)^{-1} = I.
$$

Equivalently, we could have taken $U = (UX^{-1}) \cdot X$, with $X^2 = U^TU$.