In reply to JoJo.

As long as the original data are not rank-deficient, there is an invertible linear transformation between the original basis and the (full) orthogonal PCA basis. If you know the variances of the estimates and the covariance of the betas, you can push that information forward to compute the variances of the estimates for the PCA coefficients. See the explanation by whuber on StackExchange that begins "I've been thinking about this..."