In this paper, the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered. A class of invariant asymptotic tests based on the sample covariance matrix is derived. Tests in this class are shown to be consistent and their local power functions are given. The arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix. The results are shown to generalize when the procedures are based on any affine-invariant $M$-estimate of scatter and when the population is elliptical.
"A Class of Asymptotic Tests for Principal Component Vectors." Ann. Statist. 11 (4) 1243 - 1250, December, 1983. https://doi.org/10.1214/aos/1176346337