Computing PageRank Scores

The PageRank vector π, where the ith entry is the PageRank score for webpage i, is an eigenvector for the eigenvalue λ=1 of G.
Let A=(ajk) be an n×n matrix and X a column vector. For the equation AX=λX, the roots of this equation are called eigenvalues of the matrix A. Corresponding to each eigenvalue, there will be a solution X≠0, which is called an eigenvector for the eigenvalue.
For small Google matrices, we can quickly find exact solutions to the eigensystem, πG=π. The Google matrix for the entire Web has more than 25 billion rows and columns, so computing the exact solution requires extensive time and computing resources. The oldest and easiest technique for approximating an eigenvector of a matrix is the power method:
Given a starting vector π(0), e.g., π(0)=v, the power method calculates successive iterates 
     π(k) = π(k-1) G,     where k=1,2,...,
until some convergence criterion is satisfied.
For example, the PageRank vector of the Model 1 is found as follows: 
                                           3/80  71/80   3/80   3/80
   π(0)G = vG = [ 1/4  1/4  1/4  1/4 ] × [  3/80   3/80  71/80   3/80 ]
                                          37/80   3/80   3/80  37/80
                                           1/4    1/4    1/4    1/4
 = [ 0.197  0.303  0.303  0.197 ] = π(1)

                                              3/80  71/80   3/80   3/80
   π(1)G = [ 0.197  0.303  0.303  0.197 ] × [  3/80   3/80  71/80   3/80 ]
                                             37/80   3/80   3/80  37/80
                                              1/4    1/4    1/4    1/4 
 = [ 0.208  0.247  0.341  0.208 ] = π(2)

       ...

 π(k) = [ 0.21  0.26  0.31  0.21 ] ≈ π (PageRank vector)