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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?

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