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