Given a (traceless) complex matrix $A$, I can decompose it into the sum of a diagonal matrix and a strictly upper triangular matrix via the Schur decomposition
$$ A = U(D+T)U^\dagger $$
where $U\in U(N)$, $D$ is diagonal and $T$ a strictly upper triangular matrix.
Now given $A$, I can construct a Hermitian matrix $B = AA^\dagger$ which necessarily has real eigenvalues and can be reduced to a diagonal matrix via another unitary transformation
$$ B = V \Delta V^\dagger$$
where $V\in U(N)$ and $\Delta$ is also diagonal.
My question is: Can I relate the unitary matrix that gives me the Schur decomposition to the unitary matrix that gives me the diagonal matrix, or is the relationship more complicated?