Why is the PageRank vector also the eigenvector of the web adjacency matrix? From wikipedia:


  The PageRank values are the entries of the dominant eigenvector of the
  modified adjacency matrix. This makes PageRank a particularly elegant
  metric


Can anyone please elaborate on the connection between the eigenvector and the PR vector? Why are they related?
PageRank is the stationary probability (i.e. dominant eigenvector) of the following random walk: with probability 0.85, choose a random outgoing link; with probability 0.15, choose a random web page. The "modified adjacency matrix" that Wikipedia talks about is obtained from the actual adjacency matrix $A$ by computing $0.85 A + 0.15 J$, where $J$ is the all-ones matrix. See this talk (page 13).