Selecting subset of variables most associated with the principal components of the data I have a large data matrix that I'm trying to reduce to a reasonably sized basis set. The original matrix is 916x225, and I need to reduce the number of variables (its columns) to around 50, but I want to select those that are the most representative of the complete matrix.
Specifically, I want to find a subset S of size - say - 50 variables from all, which leave the least unexplained variance in a regression of all the other variables on S ("most representative").
My current approach is to perform PCA (prcomp in R), and get the individual columns that are most associated with each principal component.  I assume that the original variable with the largest absolute value for its loading  (i.e., the largest absolute value in the rotation matrix for each variable), is thus most representative or most correlated with that PC.  
Am I interpreting this correctly?  If not, any additional guidance is appreciated.
Update:  From the comments below, I wanted to add this clarifying point in order to help focus any discussion on my intent.  I apologize that I did not convey it well in the original question.
Essentially I'm looking for a subset S of size - say - L=50 variables from all, which leave the least unexplained variance in a regression of the other variables on S ("most representative"). My hope was that by using PCA, I could find how many PCs are need for, say, 90% of the variance, then choose the variables that are most correlated with each PC. 
I thought of brute force search, too, but haven't tried that since I have 225 variables in my original matrix, and 225 choose 50 comes to about 3*e+50. That might take a very long time to compute all those linear models. 
 A: You're interpreting PCA incorrectly, I believe. The output of PCA is an orthogonal basis set of all the data observations, where the first principal component contributes the most variance, the second PC the second most, and so on so forth.
However, I'll take 'finding the column vectors most correlated with each PC' to mean 'I want to reduce my dimensionality/number of features. Selecting the variables most correlated with the PCs will not do this. But, there is a way! You can transform your observation matrix to the new basis found by your PCA, achieving just this. This is done by $T_{L} = X~W_{L}$, where $X$ is your observation matrix, $W$ is the loading matrix, $T$ is the transformed matrix, and $L$ is the number of principal components you want to keep. So, if you let $L=225$ (the maximum L, there cannot be more PCs than the original number of features), then nothing has changed. But, if you let $L=50$, then you're effectively projecting your data onto the 50th-dimension hyperplane, in the new principal component feature space. This is key! You are not SELECTING 50 features, but projecting your data onto the 50 most important PCs. So, your features will be different from the original ones, but they will be 50 features that are linear combinations (/projections) of the old ones.
Of course, there's also the fact that you have no idea if your dataset is linearly separable; you might want to check that out first before pursuing PCA.
