# How to compute $\text{trace}((A+D)^{-1}A)$

Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute

$$\text{trace}((A+D)^{-1}A)$$

Or is there a good approximation?

-

If $A$ is invertible and $\|D\|$ is small, $(A+D)^{-1}A=(I+A^{-1}D)^{-1}\approx I-A^{-1}D$ and hence $\operatorname{trace}\left((A+D)^{-1}A\right)\approx n-\operatorname{trace}(A^{-1}D)$.
I was just about to say the same with one small addition. For the last term, I think, you can derive some estimate $|\operatorname{Tr}(A^{-1}D)|=|A^{-T}:D|\leq\|A^{-1}\|_F\|D\|_F$, where $:$ is the Frobenius inner product and $\|\cdot\|_F$ is the Frobenius norm. –  Elmar Zander Mar 9 '13 at 9:10
@user1551, I am just curious. What does it mean when you put $\approx$ between two matrices? –  Easy Mar 9 '13 at 11:14