Is there such thing as a case-weighted PCA? Say I have 300 samples from a population containing two groups, A and B, and data for several variables. I have 150 from Group A and 150 from Group B. However, I know that Group A makes up roughly 20% of the population and group B makes up 80% and the two groups differ on the variables in question.
Is there a way to weight the PCA by cases to make it more representative of the population?
Would it be enough to just to do a weighted standardization?
 A: (Converting my comment into an answer so that this doesn't stay officially unanswered.)
I don't know of such a thing, but it may exist.  However, it seems to me that this isn't really much of a problem.  PCA is more of a descriptive technique than an inferential technique.  We can contrast it with running a simple product moment correlation.  If you have two variables, $X$ & $Y$, and you duplicated your data (such that you had two copies of every observation), the computed $r_{XY\ (2N)}$ wouldn't change relative to computing $r_{XY}$ on only the original $N$ rows.  What would happen is that the computed confidence interval around $r_{XY\ (2N)}$ would be too narrow, and the $p$-value would be too low.  These effects occur because Pearson's $r$ can be seen as both a descriptive statistic and an inferential statistic.  PCA doesn't really have that latter inferential attribute.  As a result, there is no harm in duplicating your data and running PCA—you should get the same eigenvectors and eigenvalues.  The implication, therefore, is that you can get a weighted PCA manually by duplicating the $n_A$ rows and copying the $n_B$ rows seven times over such that your final dataset is $2\times n_A +8\times n_B$.  Then run PCA on the enlarged dataset.  
