Eigenvalues of symmetric part of $(A+B)^{-1} B$ I want to analyze the equation $z^TMz$ where $M = (A+B)^{-1}B$, and where $A$ is symmetric positive definite, and $B$ is symmetric positive semi-definite, and compare $z^TMz$ to just $z^T z$.
I want to say something about how the number $z^TMz$ may scale when I add some positive semi-definite matrix $C$ to either $A$ or $B$  (this happens when I add data to my estimation problem, where all of this comes from).
Due to the specific form of the equation, it suffices to look at the symmetric part of $M$, namely $$G := \frac{1}{2}(M^T+ M) = \frac{1}{2} \left(B(A + B)^{-1} + (A + B)^{-1}B \right) $$
Thus, saying something about the eigenvalues of $G$ will help.
My questions are:

*

*Can we characterize (e.g. bound from above/below) the eigenvalues of $G$ somehow from $A, B$?

*What is the difference in this characterization when we add $C$ to $A$ or $B$?


Adding the matrix $C$ to $A$ results in
$$G_A = \frac{1}{2} \left(B(A + B + C)^{-1} + (A + B + C)^{-1}B \right) $$
while adding some (possibly different) $\bar C$ to $B$ gives
$$G_B = \frac{1}{2} \left((B+\bar C)(A + B + \bar C)^{-1} + (A + B + \bar C)^{-1}(B+ \bar C) \right) $$
From these equations it seems that an answer to the first question may assist the second one.
We know that $G, G_A, G_B$ will have real eigenvalues, and I expect that there can be both negative and positive eigenvalues in all problems, still the values seem to shrink as $C$ is added, but more so when it is added to $A$ rather than $B$.
 A: Via the field of values I was able to characterize some useful properties of the equation $z^T M z$ with this particular definition.
The field of values is a set defined as $F(A) = \{ z^T A z | z^Tz = 1\}$ (here I do not need to work with complex stuff).
The following will just give approximations, but that is nice.
Regarding comparing $z^T M z$ to $z^T z$, this was rather simple (to approximate), since $eig(I - A) = 1 - eig(A)$.
Due to the Rayleigh-Ritz theorem, it suffices (for my application) to look at the eigenvalues of $G$.
For a symmetric matrix $A$, then $F(A) = [\min eig(A), \max eig(A)]$ (that's the Rayleigh-Ritz theorem).
From theorems in the paper cited by @platypus we have that
$$
eig((A+B)^{-1} B) \in \frac{F(B)}{F(A) + F(B)}
$$
where set arithmetic is defined elementwise.
This means that
$$
\max eig((A+B)^{-1} B) \leq \frac{\max eig(B)}{\min eig(A) + \min eig(B)}
$$
and
$$
\min eig((A+B)^{-1} B) \geq \frac{\min eig(B)}{\max eig(A) + \max eig(B)}
$$
Then it is quite simple to see what possibly happens if we add some positive semi-definite matrix $C$ to either $A$ or $B$. However, it depends on how the eigenspace of $C$ relates to the matrix it is added to.
