I'm trying to run PCA on sample covariance matrices of various sizes (ranging between 20 x 20 to 4000 x 4000). Assume the data follows a joint multivariate normal distribution.
While derivations are great, I'm asking from an applied perspective. Bonus points for easy-to-understand papers and packages in R that help with the following:
Is there a way to test the statistical significance of the eigenvalues? Or assign a probability of a given eigenvalue occurring?
How does the relative magnitude of the eigenvalues change as we scale up the size of the covariance matrix? E.g. Let's say the first 3 principal components explain ~80% of the variance for a 20x20 matrix (not sure if this is true in practice). If we were to scale this up to a 4000x4000 matrix, would we still expect to see ~80% of the variance explained by the first 3 PCs?
Assume the population covariance matrix is not diagonal and is of full rank. Thanks!