Let $A$ be a symmetric PSD matrix, $0\leq \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_n $ are the eigenvalues. $v_i$ are the corresponding eigenvectors.
Derive a submatrix $B$ by eliminating the jth row and column from $A$. How to calculate (or approximate) the eigenvectors/eigenvalues of $B$ from that of $A$.
Cauchy interlacing theorem separates the eigenvalues of $A$ and that of $B$: $$ 0\leq \lambda_1 \leq \mu_1 \leq \lambda_2 \leq \mu_2 \leq \dots \leq \mu_{n-2} \leq \lambda_{n-1} \leq \mu_{n-1} \leq \lambda_n $$ where $\mu_1 \leq \mu_2 \leq \dots \leq \mu_{n-1} $ are $B$'s eigenvalues.
Schur complement seems relevant. PB Denton's eigenvectors from eigenvalues says
$$ |v_{i,j}|^2 \Pi_{k=1;k\neq i}^n ( \lambda_i -\lambda_k)= \Pi_{k=1}^{n-1} ( \lambda_i -\mu_k) $$
But I am not able to put pieces together: how to compute/approximate the submatrix's ith eigenvectors/eigenvalues from that of the full matrix?