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For a symmetric matrix $A$, the rationale for the success of applying QR to compute the spectral decomposition of $A = UDU^T$ is, for large $k$, the QR factorization of $A^k = Q_kR_k$ obeys, \begin{equation} Q_k\approx U. \end{equation} Could anyone help to justify the approximation?

Thanks, Jack

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    $\begingroup$ QR is interpreted today as simultaneous subspace iteration (a generalization of the power method to subspaces) on a flag of Krylov subspaces. I suggest you to take a look at a book such as Watkins' The matrix eigenvalue problem. $\endgroup$ Oct 9 '13 at 17:37

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