Autonne-Takagi factorization for complex symmetric matrices

Question

Let us consider a complex symmetric matrix $\Omega$. The Autonne-Takagi factorization tells us that a unitary matrx $U$ exists such that $$U^T\Omega U = D,$$ where $D$ is a real diagonal matrix with nonnegative entries. I am trying to prove, from this expression, that $D$ is the square root of the diagonal form of the matrix $\Omega^\dagger\Omega$, diagonalized by the matrix $U$. In the specific, that $$ U^\dagger\Omega^\dagger\Omega U = D^2.$$ How can I derive the second expression from the first one? Although it seems pretty simple, I may be missing some obvious property.

Answer 1

$D^2=D^\ast D=(U^T\Omega U)^\ast (U^T\Omega U)=U^\ast\Omega^\ast\Omega U$.