# The rationale of QR algorithm for computing eigenvectors

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, $$Q_k\approx U.$$ Could anyone help to justify the approximation?

Thanks, Jack

• 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. Oct 9 '13 at 17:37