Assume A is an NXN matrix, diagonalizable, with eigenvalues and associated eigenvectors . The power method computes
The argument for why this process converges follows easily if we think of the initial guess . Then
This argument shows that the iterations approximate the true eigenvalue with an error of . This is the same error (in norm) for the approximation of the true eigenvector by .
If two eigenvalues are close in absolute value, the power iteration will converge slowly. In this case, it may be advantageous to shift the eigenvalues. In other words, perform the power method on a matrix , with chosen to speed up the calculation. The eigenvalue you compute is, of course, shifted also.
Another method for speeding up the calculation is Aitken's extrapolation.
Finally, to obtain other eigenvalues (say the second) FOR SYMMETRIC MATRICES is to apply the power method to the matrix where is the normlized eigenvector. You should convince yourself that has the same eigenvectors as A, and the same eigenvalues, except that has been replaced by 0.