Here's a useful PCA paper I found while writing HW1's renderer

I used this technique in a real-time GPU DXT1 encoder I wrote around 10 years ago:

"Candid Covariance-Free Incremental Principal Component Analysis"
http://www.cse.msu.edu/~weng/research/CCIPCApami.pdf

With this approach you can compute a decent-enough PCA in a few lines of shader code.

HW1 used this encoder to compress all of the GPU splatted terrain textures into a GPU texture cache. One of my coworkers, Colt McAnlis, designed and wrote the game's amazing terrain texture caching system.