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?
2026-04-02 09:53:14.1775123594
How to compute $\text{trace}((A+D)^{-1}A)$
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If $A$ is invertible and $\|D\|$ is small, $(A+D)^{-1}A=(I+A^{-1}D)^{-1}\approx I-A^{-1}D$ and hence $\operatorname{trace}\left((A+D)^{-1}A\right)\approx n-\operatorname{trace}(A^{-1}D)$.