  Consider a data matrix $Y = [\mathbf{y}_1, \cdots, \mathbf{y}_N]$ of size $M
\times N$, where the columns are independent observations from a random vector
$\mathbf{y}$ with zero mean and population covariance $\Sigma$. Let
$\mathbf{u}_i$ and $\mathbf{v}_j$ denote the left and right singular vectors of
$Y$, respectively. This study investigates the eigenvector/singular vector
overlaps $\langle {\mathbf{u}_i, D_1 \mathbf{u}_j} \rangle$, $\langle
{\mathbf{v}_i, D_2 \mathbf{v}_j} \rangle$ and $\langle {\mathbf{u}_i, D_3
\mathbf{v}_j} \rangle$, where $D_k$ are general deterministic matrices with
bounded operator norms. We establish the convergence in probability of these
eigenvector overlaps toward their deterministic counterparts with explicit
convergence rates, when the dimension $M$ scales proportionally with the sample
size $N$. Building on these findings, we offer a more precise characterization
of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the
population covariance $\Sigma$.
