PageRank computes its scores by multiplying a vector by the link matrix over and over until it stops changing. Why does that converge to the answer?

B. Because repeated multiplication amplifies the dominant eigenvector

Each multiplication scales every eigen-component by its eigenvalue, so the largest dominates and the rest fade, the fixed point is the principal eigenvector. Power iteration is both the proof the circular definition settles and the algorithm for computing it at web scale, with no eigen-solver required.