Page "QR algorithm" Paragraph 11

The QR algorithm can be seen as a more sophisticated variation of the basic " power " eigenvalue algorithm.

Recall that the power algorithm repeatedly multiplies A times a single vector, normalizing after each iteration.

The vector converges to an eigenvector of the largest eigenvalue.

Instead, the QR algorithm works with a complete basis of vectors, using QR decomposition to renormalize ( and orthogonalize ).

For a symmetric matrix A, upon convergence, AQ = QΛ, where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of all the orthogonal similarity transforms required to get there.

Thus the columns of Q are the eigenvectors.