The joint density of top k sample eigenvectors and principal subspace inference

Abstract

In this paper, the exact joint density of the top $k$ sample eigenvectors is derived for any $p\times p$ population covariance matrix $\Sigma$, sample size $n>p-1$, and $1\le k\le p$. Using symmetric functions such as zonal polynomials and a dual summation identity by I. G. Macdonald, the result is expressed as a series in terms of determinants of differential operators. A matrix Kummer transformation converts the resulting local alternating expansion into a globally absolutely convergent series over the entire positive definite matrix space. For $k=2$, the ordered eigenvalue integrals reduce to the Gauss hypergeometric function ${}_2{F}_1$ with a simple closed form when $n=p+1$. Previously, explicit formulas were only available for a scalar matrix $\Sigma ={\sigma }^2{I}_p$ or a general matrix $\Sigma$ with $p=2$ or $k=1$, obtained by T. W. Anderson and T. Sugiyama, respectively. The frame law also induces an exact Grassmann density that provides a finite-sample benchmark for principal subspace inference.