In the paper under review the author studies two models of deformations of random matrices.

1.

In the first one, the following Wigner matrix: \[ X_n=\begin{bmatrix} X_{11}^{(n)} & X_{12}^{(n)} & \cdots & X_{1n}^{(n)} \\ X_{12}^{(n)} & X_{22}^{(n)} & \cdots & X_{2n}^{(n)} \\ \vdots & \vdots & \ddots & \vdots \\ X_{1n}^{(n)} & X_{2n}^{(n)} & \cdots & X_{nn}^{(n)} \\ \end{bmatrix} \] and the perturbation model: \[ W_n:=\frac 1{\sqrt n}X_n+A_n, \] where \(A_n\) is a deterministic matrix or random matrix independent of \(X_n\), with eigenvalues \(\gamma_1^{(n)}\), \(\ldots\), \(\gamma_n^{(n)}\), is considered.

2.

In the second one, the following Wishart matrix: \[ X_n=\begin{bmatrix} X_{11}^{(n)} & X_{12}^{(n)} & \cdots & X_{1m}^{(n)} \\ X_{12}^{(n)} & X_{22}^{(n)} & \cdots & X_{2m}^{(n)} \\ \vdots & \vdots & \ddots & \vdots \\ X_{1n}^{(n)} & X_{2n}^{(n)} & \cdots & X_{nm}^{(n)} \\ \end{bmatrix} \] and the perturbation model: \[ S_n:=\frac 1n\Sigma_n^{\frac 12}X_nX_n^T\Sigma_n^{\frac 12}, \] where \(\Sigma_n\) is a covariance matrix, is taken into consideration.

For the both cases the author studies the limiting spectral measure in the direction of an eigenvector of the perturbation. From the obtained results he derives two applications: one connected to the value of the outlier and the norm its associated eigenvector projection in the direction of the spike, and one connected to the properties of the projection of non-outlier eigenvectors in the direction of the spike.