We believe that the application of PCA to genetic data—and our way of analyzing the data—provides a natural method of uncovering population structure, for reasons that are subtle; thus, it is useful to spell them out explicitly. In most applications of PCA, the multivariate data has an unknown covariance, and PCA is attempting to choose a subspace on which to project the data that captures most of the relevant information. In many such applications, a formal test for whether the true covariance is the identity matrix makes little sense.
