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Let $A \in \mathbb C^{d \times d}$. The Schur factorization of $A$ is the decomposition $A = U T U^\ast$, where $T \in \mathbb C^{d \times d}$ is an upper triangular matrix and $U \in \mathbb C^{d \times d}$ is unitary.

If $A$ is symmetric, then it follows immediately that $T$ is diaogonal. This means, the spectral decomposition of symmetric matrices can be reduced to the Schur factorization of general matrices.

I am wondering about opposite relations. Can I reduce the Schur factorization of general matrices to the spectral decomposition of symmetric matrices?

To illustrate the point, if I can compute the singular value decomposition of any symmetric matrix $S$, then I can compute the SVD of any other matrix $A$ by computing the SVD of $S = A^t A$ and some additional matrix-vector multiplications. I wonder whether anything analogous can be shown for the Schur decomposition.

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