Given square matrices $A$, $B$, if there exists a symmetric matrix $X$ with entries in $(0,1)$ such that $$ AXA^T = BXB^T, $$ does that imply $A = B$ ?
What about a specific condition, e.g. when $A, B$ are symmetric?
Sizes of $A, B$ necessarily have to be the same for the equation to be valid.