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

https://math.stackexchange.com/questions/3869/what-is-the-intuitive-relationship-between-svd-and-pca