# Schur Decomposition of a complex matrix and spectral decomposition

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?