Why is the PageRank vector also the eigenvector of the web adjacency matrix?
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).