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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?

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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).

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