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been replaced by 0.