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.



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


