PCA is map the data to lower dimensional. In order for PCA to do that it should calculate and rank the importance of features/dimensions.
There are 2 ways to do so.
- using eigenvalue and eigenvector in covariance matrix to calculate and rank the importance of features
- using SVD on covariance matrix to calculate and rank the importance of the features
SVD (covariance matrix) = [U S V']
after ranking the features/ dimensions then it will choose the most important ones (k) and map the actual data to k dimension.
in case PCA used SVD to rank the importance of features, then U matrix will have all features ranked, we choose the first k columns which represent the most important one.
to determinate k we can use S matrix.
This is a link that explain to you why PCA can use SVD instead of eigvector/eignvalue