What is the right citation for the power iteration method to find eigenvalues, if I want to cite the method in a paper? I've seen some Google PageRank references in this context. But Brin and Page didn't invent the power iteration method, did they?

Several researchers indicate that the power iteration method dates back to Herman Müntz. According to the survey aricle "Eigenvalue computation in the 20th century" by Golub and van der Vorst,
Ortiz and Pinkus also mention in "Herman Müntz: A Mathematician’s Odyssey" that
The relevant papers by Müntz:



I vaguely remember, that this is due to Hotelling (1933). I don't have the monography of S. Mulaik "The foundations of factor analysis" (1972) at hand, but googlebooks let's you have a peek for "Hotelling". At page 114/115 this is discussed and Mulaik writes: "However, in 1933 Hotelling published a paper in the Journal of Educational Psychology which described a method of finding the characteristic equation directly (...)" Page 115: "(...) permitting us thereby to pick from among the eigenvectors the one associated with the largest eigenvalue. As a matter of fact, both kinds of methods are available. The first, which converges to the eigenvector associated with the largest eigenvalue, is due to Hotelling.(...)" The method named after Jacobi to find the eigenvalues/eigenvectors is the special case of rotating the columns of a matrix to approximate a certain maximization criterion iteratively. This can be done if a symmetric matrix was decomposed for instance in its two triangular choleskyfactors, and the lower triangular factor is ("Jacobi") rotated to "principal components position". Here all pairs of columns are rotated to maximize some criterion and this is repeated until some convergence criterion is satisfied (I can provide that criterion if needed because I've implemented it in a software). The book of S. Mulaik is a bit aged and of the year 1972, and although there is a lot of modern development in factoranalysis I rate it as still the best monography/standard textbook for the basic understanding of factor analysis and related basic methods of linear algebra (as well for the history...) 

