  We sketch the history of spectral ranking, a general umbrella name for
techniques that apply the theory of linear maps (in particular, eigenvalues and
eigenvectors) to matrices that do not represent geometric transformations, but
rather some kind of relationship between entities. Albeit recently made famous
by the ample press coverage of Google's PageRank algorithm, spectral ranking
was devised more than a century ago, and has been studied in tournament
ranking, psychology, social sciences, bibliometrics, economy and choice theory.
We describe the contribution given by previous scholars in precise and modern
mathematical terms: along the way, we show how to express in a general way
damped rankings, such as Katz's index, as dominant eigenvectors of perturbed
matrices, and then use results on the Drazin inverse to go back to the dominant
eigenvectors by a limit process. The result suggests a regularized definition
of spectral ranking that yields for a general matrix a unique vector depending
on a boundary condition.
