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

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.

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.