I'm having difficulty solving a linear algebra problem:
Let $A,B,C,D$ be real $n \times n$ matrices. Show that there is a non-zero $n \times n$ matrix $X$ such that $AXB$ and $CXD$ are both symmetric.
There is an accompanying hint:
Show that the set of all matrices $X$ for which $AXB$ is symmetric is a vector space, and compute its dimension.
I introduced some notation: let $S(A,B)$ denote the set of all $n \times n$ real matrices such that $AXB$ is symmetric. It's easy to prove that for any $A,B$ real $n \times n$ matrices $S(A,B)$ is a subspace of $\cal {M}_{n \times n} (\mathbb{R})$-it's closed under addition, scalar multiplication, and contains the zero matrix. Now I'm not entirely sure where to go. I'd like to be able to show that the space $S(A,B) \cap S (C,D)$ has dimension greater than $0$, but I haven't made any progress thus far.
Any help would be greatly appreciated!