What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?

Question

I have two problems related to eigenvalues of negative definite matrices:

1. I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[M_{ij}]$. Now I want to derive the relation between the eigenvalues of $M$ and $S$. I am particularly looking for a relation between the minimum and maximum eigenvalues.

2. Now if $M=A^T+A\prec 0$ then what can be said about the relationship between the eigenvalues of $M$ and $A$. I have tried that if the condition of symmetry is not imposed then $A, A^T \prec 0$ with the real part of all eigenvalues negative. But if this is so, can we have any eigenvalue inequality on real parts only.

Answer 1

As for your first question, $S^{-1}$ is a diagonal block of $M^{-1}$. You can only say that its eigenvalues are interlaced with those of $M^{-1}$. Since all the eigenvalues of $M$ and $S$ are negative, it amounts to saying that the eigenvalues of $S$ are interlaced with those of $M$~: $$\lambda_1\le\mu_1\le\lambda_2\le\cdots\le\mu_{n-1}\le\lambda_n.$$

As for your second question, see my previous answer https://mathoverflow.net/q/52588.