I'm looking to solve for a specific $X$ in the following equation: $$X^T\Sigma X = D,$$ where $\Sigma \succ 0$, $D$ is a diagonal matrix with strictly positive entries, and all matrices are square. It may also be worth noting that $\det(\Sigma) = \det(D)$.
I'm looking for a (known to exist) solution $X$ which can be written as $I-W$, where $W$ is permutation-similar to a strictly-triangular matrix. Using Sylvester's Law of Inertia, I'm able to solve for an $X$ by finding the transformation which converts $\Sigma$ and $D$ to the same diagonal matrix (since they both have all positive entries). However, I don't know how to take this solution to one that matches the form I-W as described above.
Alternatively, I've also considered how one might solve this with the solution to a convex program, but haven't had much success so far.