Additive/Multiplicative Free Subordination Property and Limiting Eigenvectors of Spiked Additive Deformations of Wigner Matrices and Spiked Sample Covariance Matrices

When some eigenvalues of a spiked additive deformation of a Wigner matrix or a spiked multiplicative deformation of a Wishart matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the perturbation. We point out that the subordination function relative to the free (additive or multiplicative) convolution plays an important part in the asymptotic behavior.