For Gaussian random matrices, the squared overlaps between singular vectors of a submatrix and of the full matrix have explicit limiting Cauchy-like formulas in the Marchenko-Pastur regime.
Diffusions of perturbed principal component analysis.Journal of mul- tivariate analysis, 29(1):127–136, 1989
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Eigenvector Overlaps of Random Covariance Matrices and their Submatrices
For Gaussian random matrices, the squared overlaps between singular vectors of a submatrix and of the full matrix have explicit limiting Cauchy-like formulas in the Marchenko-Pastur regime.