Yes, there may be a reason to withdraw from rotation in factor analysis. That reason is actually similar to why we usually do not rotate principal components in PCA (i.e. when we use it primarily for dimensionality reduction and not to model latent traits).
After extraction, factors (or components) are orthogonal$^1$ and are usually output in descending order of their variances (column sum-of-squares of the loadings). The 1st factor thus dominates. Junior factors statistically explain what the 1st one leaves unexplained. Often that factor loads quite highly on all the variables, and that means that it is responsible for the background correlatedness among the variables. Such 1st factor is sometimes called general factor or g-factor. It is considered responsible for the fact that positive correlations prevail in psychometrics.
If you are interested in exploring that factor rather than disregard it and let it dissolve behind the simple structure, don't rotate the extracted factors. You may even partial out the effect of general factor from the correlations and proceed to factor-analyze the residual correlations.
$^1$ The difference between extraction factor/component solution, on one hand, and that solution after its rotation (orthogonal or oblique), on the other hand, is that - the extracted loading matrix $\bf A$ has orthogonal (or nearly orthogonal, for some methods of extraction) columns: $\bf A'A$ is diagonal; in other words, the loadings reside in the "principle axis structure". After rotation - even a rotation preserving orthogonality of factors/components, such as varimax - the orthogonality of loadings is lost: "principle axis structure" is abandoned for "simple structure". Principal axis structure allows to sort out among the factors/components as "more principal" or "less principal" (and the 1st column of $\bf A$ being the most general component of all), while in simple structure equal importance of all the rotated factors/components is assumed - logically speaking, you cannot select them after the rotation: accept all of them (Pt 2 here). See picture here displaying loadings before rotation and after varimax rotation.
The two citations do not generally contradict each other and both look to me correct. The only underwork is in Perhaps you mean sum of squared loadings for a principal component, after rotation one should better drop word "principal" since rotated components or factors are not "principal" anymore, to be rigorous. Also (important!) the second citation is correct only when "factor analysis" is actually PCA method (like it is in SPSS by default) and so factors are just principal components. But the table you present is not after PCA, and I wonder whether they are from the same text and wasn't there some misprint.
In the extraction summary table you display there was 23 variables analyzed. Eigenvalues of their correlation matrix are shown in the left section "Initial eigenvalues". No factors have been extracted yet. These eigenvalues correspond to the variances of Principal components (i.e. PCA was performed), not of factors. Adjective "initial" means "at the initiation point of the analysis" and does not imply that there must be some "final" eigenvalues.
The (default in SPSS) Kaiser rule "eigenvalues>1" was used to decide how many factors to extract, so, 4 factors will come. The "eigenvalues>1" rule is based on PCA's eigenvalues (i.e. the eigenvalues of the intact, input correlation matrix).
Extraction of them was done by Principal axis method and the matrix of loadings obtained. Sums of squared loadings in the matrix columns are the factors' variances after extraction. These values appear in the middle section of your table.
These numbers, generally, should not be called eigenvalues because factor extractions not necessarily are based right on the eigendecomposition of the input data - they are specific algorithms on their own. Even Principal axis method which does involve eigenvalues deal with eigenvalues of a repeatedly "trained" matrix, not an original correlation matrix.
But if you had been doing PCA instead of FA then the 4 numbers in the middle column would have been the 4 first eigenvalues identical to the 4 largest ones on the left: in PCA, no fitting take place and the extracted "latent variables" are the PCs themselves, which eigenvalues are their variances.
In the right section, sums of squared loading after rotaion of the factors are shown. The variances of these new, rotated factors. Please read more about rotated factors (or components), especially footnote 4, and that they are neither "principal" anymore nor this-one-to-that-one correspondent to the extracted ones. After rotation, "2nd" factor, for example, is not "2nd extracted factor, rotated". And it also could have greater variance than the "1st" one.
So,
- No, you can't speak of eigenvalues after rotation. No matter be it
orthogonal or oblique.
- You can't even say - at least should better
avoid - of eigenvalues even after extraction of factors unless
these factors are principal components$^1$. (An instructive example showing confusion similar to yours with ML factor extraction.) Variances of factors are SS loadings, not eigenvalues, generally.
- Rotated factors don't correspond one-to-one to the extracted ones.
- The % of total variation
explained by the factors is 40.477% in your example, not 50.317%.
The first number is less because FA factors explain all the assumed
common variation which is less than the portion of total
variation skimmed by the same number of PCs. May say in your report, "The 4-factor solution is responsible for the common variance constituting 40.5% of the total variance; while 4 principal components would account for 50.3% of the total variance".
$^1$ (Before factor rotation) variances of factors (pr. components) are the eigenvalues of the correlation/covariance matrix of the data if the FA is PCA method; variances of factors are the eigenvalues of the reduced correlation/covariance matrix with final communalities on the diagonal, if the FA is PAF method of extraction; variances of factors do not correspond to eigenvalues of correlation/covariance matrix in other FA methods such as ML, ULS, GLS (see). In all cases, variances of orthogonal factors are the SS of the extracted/rotated - final - loadings.
Best Answer
Reason for rotation. Rotations are done for the sake of interpretation of the extracted factors in factor analysis (or components in PCA, if you venture to use PCA as a factor analytic technique). You are right when you describe your understanding. Rotation is done in the pursuit of some structure of the loading matrix, which may be called simple structure. It is when different factors tend to load different variables $^1$. [I believe it is more correct to say that "a factor loads a variable" than "a variable loads a factor", because it is the factor that is "in" or "behind" variables to make them correlate, but you may say as you like.] In a sense, typical simple structure is where "clusters" of correlated variables show up. You then interpret a factor as the meaning which lies on the intersection of the meaning of the variables which are loaded enough by the factor; thus, to receive different meaning, factors should load variables differentially. A rule of thumb is that a factor should load decently at least 3 variables.
Consequences. Rotation does not change the position of variables relative to each other in the space of the factors, i.e. correlations between variables are being preserved. What are changed are the coordinates of the variable vectors' end-points onto the factor axes - the loadings (please search this site for "loading plot" and "biplot", for more)$^2$. After an orthogonal rotation of the loading matrix, factor variances get changed, but factors remain uncorrelated and variable communalities are preserved.
In an oblique rotation factors are allowed to lose their uncorrelatedness if that will produce a clearer "simple structure". However, interpretation of correlated factors is a more difficult art because you have to derive meaning from one factor so that it does not contaminate the meaning of another one that it correlates with. That implies that you have to interpret factors, let us say, in parallel, and not one by one. Oblique rotation leaves you with two matrices of loadings instead of one: pattern matrix $\bf P$ and structure matrix $\bf S$. ($\bf S=PC$, where $\bf C$ is the matrix of correlations between the factors; $\bf C=Q'Q$, where $\bf Q$ is the matrix of oblique rotation: $\bf S=AQ$, where $\bf A$ was the loading matrix prior any rotation.) The pattern matrix is the matrix of regressional weights by which factors predict variables, while the structure matrix is the correlations (or covariances) between factors and variables. Most of the time we interpret factors by pattern loadings because these coefficients represent the unique individual investment of the factor in a variable. Oblique rotation preserves variable communalities, but the communalities are no longer equal to the row sums of squares in $\bf P$ or in $\bf S$. Moreover, because factors correlate, their variances partly superimpose$^3$.
Both orthogonal and oblique rotations, of course, affect factor/component scores which you might want to compute (please search "factor scores" on this site). Rotation, in effect, gives you other factors than those factors you had just after the extraction$^4$. They inherit their predictive power (for the variables and their correlations) but they will get different substantial meaning from you. After rotation, you may not say "this factor is more important than that one" because they were rotated vis-a-vis each other (to be honest, in FA, unlike PCA, you may hardly say it even after the extraction because factors are modelled as already "important").
Should one prefer orthogonal or oblique rotation? Well, orthogonal factors are easier to interpret and the whole factor model is statistically simpler (orthogonal predictors, of course). But there you impose orthogonality on the latent traits you want to discover; are you sure they should be uncorrelated in the field you study? What if they are not? Oblique rotation methods$^5$ (albeit each having their own inclinations) allow, but don't force, factors to correlate, and are thus less restrictive. If oblique rotation shows that factors are only weakly correlated, you may be confident that "in reality" it is so, and then you may turn to orthogonal rotation with good conscience. If factors, on the other hand, are very much correlated, it looks unnatural (for conceptually distinct latent traits, especially if you are developing an inventory in psychology or such, - recall that a factor is itself a univariate trait, not a batch of phenomena), and you might want to extract less factors, or alternatively to use the oblique results as the batch source to extract the so-called second-order factors.
$^1$ Thurstone brought forward five ideal conditions of simple structure. The three most important are: (1) each variable must have at least one near-zero loading; (2) each factor must have near-zero loadings for at least m variables (m is the number of factors); (3) for each pair of factors, there are at least m variables with loadings near zero for one of them, and far enough from zero for the other. Consequently, for each pair of factors their loading plot should ideally look something like:
This is for purely exploratory FA, while if you are doing and redoing FA to develop a questionnaire, you eventually will want to drop all points except blue ones, provided you have only two factors. If there are more than two factors, you will want the red points to become blue for some of the other factors' loading plots.
$^2$
$^3$ The variance of a factor (or component) is the sum of its squared structure loadings $\bf S$, since they are covariances/correlations between variables and (unit-scaled) factors. After oblique rotation, factors can get correlated, and so their variances intersect. Consequently, the sum of their variances, SS in $\bf S$, exceeds the overall communality explained, SS in $\bf A$. If you want to reckon after factor i only the unique "clean" portion of its variance, multiply the variance by $1-R_i^2$ of the factor's dependence on the other factors, the quantity known as anti-image. It is the reciprocal of the i-th diagonal element of $\bf C^{-1}$. The sum of the "clean" portions of the variances will be less than the overall communality explained.
$^4$ You may not say "the 1st factor/component changed in rotation in this or that way" because the 1st factor/component in the rotated loading matrix is a different factor/component than the 1st one in the unrotated loading matrix. The same ordinal number ("1st") is misleading.
$^5$ Two most important oblique methods are promax and oblimin. Promax is the oblique enhancement of varimax: the varimax-based structure is then loosed in order to meet "simple structure" to a greater degree. It is often used in confirmatory FA. Oblimin is very flexible due to its parameter gamma which, when set to 0, makes oblimin the quartimin method yielding most oblique solutions. A gamma of 1 yields the least oblique solutions, the covarimin, which is yet another varimax-based oblique method alternative to promax. All oblique methods can be direct (=primary) and indirect (=secondary) versions - see the literature. All rotations, both orthogonal and oblique, can be done with Kaiser normalization (usually) or without it. The normalization makes all variables equally important at the rotation.
Some threads for further reading:
Can there be reason not to rotate factors at all? (Check this too.)
Which matrix to interpret after oblique rotation - pattern or structure?
What do the names of factor rotation techniques (varimax, etc.) mean? (A detailed answer with formulae and peseudocode for the orthogonal factor rotation methods)
Is PCA with components rotated still PCA or is a factor analysis?