I was wondering if anybody has any suggestions on the following problem:

Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which MAXIMIZES the Frobenius norm of the commutator, i.e.

$$ R = \arg\max_{R' \in O(N)}||[R',S]||_F^2 = \arg\max_{R' \in O(N)}||R'S - SR'||_F^2, $$ where $O(N)$ is the set of all orthogonal matrices.

Clearly $||[R,S]||_F^2 \leq 2||S||_F^2$, provides an upper bound. I've found some papers which are relevant to this problem, in particular Commutators with maximal Frobenius norm, which identify matrices which satisfy this upper bound, but the resulting matrices will not be orthogonal. I'm not confident the results there are so applicable.

Any ideas, help or suggestions would be greatly appreciated.