It's proved here that every real antisymmetric matrix is orthogonally similar to its transpose?
Now let $A,B$ a pair of symmetric and antisymmetric matrices $(A^T=- A,B^T=B)$.
Is it true that $A,B$ are simultaneously orthogonally similar to their transposes? i.e Does there exists a matrix $S \in O(n)$ such that
By this paper, it seems the answer is positive. However, the paper uses some sophisticated arguments whichI would like to avoid, and the criterion it gives for simultaneous similarity is non-trivial to check. (based on some series of trace equalities, see Corollary 2.3 there).
Is there a more elementary approach to see this?