Relation between the diagonalization of a matrix and of its sub-block

Question

I have a symmetric matrix $M\in\mathbb{R}^{k\times k}$ whose diagonalization is known: $$ M = R D R' $$ where $D$ is diagonal and $R'$ is the transpose of $R$. Now I take a squared sub-block of $M$, considering only the first $n<k$ columns and rows, $M_n$.

Is there any known result that relates the diagonalization of $M$ with the one of $M_n$?

I have the feeling that the eigenvalues of $M_n$ should be related to those of $M$.

Answer 1

By the min-max principle, if the eigenvalues of $M$ are $\lambda_1 \le \ldots \le \lambda_k$ and the eigenvalues of $M_n$ are $\mu_1 \le \ldots \le \mu_n$ (both counted by multiplicity), then $\lambda_i \le \mu_i \le \lambda_{k-n+i}$.