Proof of existence of square root of unitary and symmetric matrix I'm struggling with this exercise
Let $U$ be a unitary and symmetric matrix ($U^T = U$ and $U^*U = I$).
Prove that there exists a complex matrix $S$ such that:


*

*$S^2 = U$

*$S$ is a unitary matrix

*$S$ is symmetric

*Each matrix that commutes with $U$, commutes with $S$


First, solve for the case of diagonal matrices, which shouldn't be too hard. Then, prove $U$ is diagonalisable, and see if you can use that result to reduce to the previous case.
