Thus, we expect that the theoretical covariance matrix (approximated by the sample covariance) will have K − 1 “large” eigenvalues, with the remainder small and reflecting the sampling variance. We call the eigenvectors of the theoretical covariance, corresponding to the large eigenvalues, “axes of variation.” These are a theoretical construct, as of course we only observe the sample covariance. Nevertheless, for eigenvectors that are highly significant by our tests, we expect the corresponding eigenvector to correlate well with the true “axis of variation.” An analogy here is that we would often think of an allele as having a “true” population frequency in a population, and would regard the frequency of the allele in a sample as an estimator of the true frequency.
