Let be the eigenvector corresponding to
. The *angle*
between and is defined by the relation

If the starting vector and the eigenvector are perpendicular to each other, then . In this case the power method does not converge in exact arithmetic. On the other hand, if , the power method generates a sequence of vectors that become increasingly parallel to . This condition on is true with very high probability if is chosen at random.

The convergence rate of the power method depends on , where is the second largest eigenvalue of in magnitude. This ratio is generally smaller than , allowing adequate convergence. But there are cases where this ratio can be very close to , causing very slow convergence. For detailed discussions on the power method, see Demmel [114, Chap. 4], Golub and Van Loan [198], and Parlett [353].