There exists $H$ such $S=XH^t+HX^t$ for every simetric $S$

Question

Given the function $f(X)=X*X^t$ then If $X^t=X^{-1}$ for every simetric matrix $S$ there exists $H$ such that $f'(X)H=S$

So i know that $f'(X)H$ is simetric regardless of $X$ being orthogonal, in fact $f'(X)H=XH^t+HX^t$ but i don't see How to prove the existence for every simetric matrix

Answer 1

Finally found the answer.

Just take $H=SX/2$ so

$XH^t+HX^t= 1/2*XX^{-1}S^t+1/2*SXX^{-1}=S$

The Idea is Just to look what Will cancel the $X$ when looking for the transposed