Does the first extracted factor in EFA always have the highest eigenvalue? I have run many EFAs (exploratory factor analysis) and always find that the first extracted factor has the highest eigenvalue.
However, I read in Petscher et al (2013) "Applied Quantitative Analysis in Education and the Social Sciences" a chapter by Rex Kline that states (p183) that "The first extracted factor tends to have the highest eigenvalue" (emphasis mine). 
As far as I can tell Kline does not explain the circumstances in which the first extracted factor does not have the highest eigenvalue. How can this happen?
 A: I have no access to the text you are citing, but let me suppose several explanations; one of them is possibly the one which fits in the context of the citation.


*

*The extraction itself needs not that the factors appear in descending order of their variances (sum of squared loadings). The important requirement is only that they are orthogonal. Even in PCA, which - unlike the majority of FA methods - hunts to maximize the variance, the eigenvalues can first appear in not decreasing order. It is afterwards that they (and the corresponding eigenvectors/loadings) are being sorted by the procedure. The same thing is in FA. Some eigen-decomposition and PCA functions in some packages do not do the sorting. However, even then the first eigenvalue is usually the highest. The principal distinction between FA and PCA in the issue we are discussing is that PCA components have to be sorted sooner or later because the purpose of the analysis is to select some number of first most strong of them; but in FA all the extracted factors are already modeled as existent (important): therefore it makes neither theoretical nor practical difference how you order them.

*Another option which comes to mind is that the authors speak about how it is after the rotation of factors. Of course, after it variances change and may go in any order, not necessarily descending one.

*One more guess is that sometimes, rarely, variances of several first (or even all) factors or components are the same or very close by magnitudes. One can easily generate such "discus" shape data clouds artificially.

