Eigenvalues of $(A+B)^{-1}$ Suppose I know the eigenvalues of $A$ and $B$. Is there a way to write eigenvalues of the following?
(1). $(A+B)$
(2). $(I+A)$
(3). $(I+A)^{-1}$
(4). $(A+B)^{-1}$
where $A, B$ are matrices in complex field and $I$ is identity matrix. Also in the cases (3) and (4) is there a way to write the inverses matrices also? 
PS. This is related to my previous question and is not homework Monotonic log det function?
Appreciate any form of insight. Thank you.
Edit I also want to write the eigenvalues of $AB^{-1}$ from $A$ and $B$. Is it possible under some assumptions? The matrices are defined in the complex field. I need this to answer another question Monotonicity of $\log \det R(d_i, d_j)$
 A: For general (diagonalisable) matrices $A,B$, their sets of eigenvalues do not determine those of $A+B$; since their eigenvectors are unrelated there is really nothing easy one can say here. If $A$ and $B$ commute, then one can say a bit more: the matrices have a basis of common eigenvectors, and the eigenvalues of $A+B$ can be found by matching up the eigenvalues of $A$ with those of $B$ in some manner, and adding up those pairs. Note that there will in genral be $n!$ different matchings, so still potentially that many different outcomes (using diagonal matrices, it is easy to see such possibilties can indeed be realised).
On the other hand eigenvectors of $A$ are also eigenvectors of $A+\mu I$ for any scalar $\mu$, and assuming invertibility also eigenvectors of $A^{-1}$; you can easily see what happens to the eigenvalues. This gives positive answers to (2) and (3).
A: If matrices $A, A^*, B, B^*$ commute, they are simultaneously diagonalizable over $\mathbb{C}$. Then you can simply assume (by a possible change of basis) that you are working with diagonal matrices. In general, there are more subtetlies like MvL points out.
