Google’s PageRank has been called “the billion dollar eigenvector”; the New York Times calls it “the key invention of Google’s founders” Larry Page and Sergey Brin. In The Book of mathematical algorithms, PageRank and the FFT decoder for the 4-by-3 telephone touchpad tone matrix must surely be in the first chapter. The Google matrix is a real nonnegative stochastic matrix, so Markov chains and Perron-Frobenius theory are a natural setting in which to analyze it. However, mathematics is rich with examples in which expanding the “natural” setting of a problem actually simplifies and clarifies the analysis. We discuss how basic methods of complex matrix analysis provide a better and more elementary context in which to understand PageRank. Page and Brin’s original paper (unpublished: not a smart step toward tenure) is available online: just Google “page brin pagerank” and a link to their 1998 Stanford technical report is the first one displayed.