How do eigenvectors and eigenvalues change when we remove a row/column pair of a matrix?

Let $A$ be a symmetric matrix, with eigenvalues $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Let $B$ be the matrix obtained by deleting the $k$-th row and column from $A$, with eigenvalues $\mu_1 \leq \mu_2 \leq \cdots \leq \mu_{n-1}$. Then $$\lambda_1 \leq \mu_1 \leq \lambda_2 \leq \mu_2 \leq \lambda_3 \leq \cdots \leq \lambda_{n-1} \leq \mu_{n-1} \leq \lambda_{n}.$$ This is a special case of Cauchy's interlacing theorem. The operator $P$ in the wikipedia article should be taken to be the projection on the coordinate vectors other than the $k$-th one.

The word you're looking for is downdating, and I cannot do better than to point out these two survey papers, and this article. I will also have to make the reminder that it makes better numerical sense to compute the singular values of $\mathbf{D}$ rather than the eigenvalues of $\mathbf{D}^T \mathbf{D}$.