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$C = A+D$, $A$ being square matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$

Edit 2: (important edit) Iam interested in this question, because my matrix $A$ is huge and so is $C$. So computing inverse of $C$ is not practical, but luckily the matrix $A$ is unitary, so $A^{-1} = A^*$, so I easily have $A^{-1}$, and finding ways to use it to get $C^{-1}$.

Rajesh D
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  • $C$ need not be even invertible. Take $A=I$ and $D=-I$. – Anurag A Oct 30 '18 at 05:47
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    @AnuragA : I am not talking about $A = I$. just a general case. Just asume $C$ is invertible, is there a way to compute it faster from knowledge of $A^{-1}$ – Rajesh D Oct 30 '18 at 05:52
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    In general also there is no guarantee that $C$ is invertible. For example, take $A=\begin{bmatrix}1&2\0&-2\end{bmatrix}$ and $D=\begin{bmatrix}-1&0\0&2\end{bmatrix}$. Then $C=\begin{bmatrix}0&2\0&0\end{bmatrix}$ is NOT invertible. – Anurag A Oct 30 '18 at 05:54

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I think that you cannot expect better than a complexity $\sim n^3$.

Indeed i) $(A+D)^{-1}=D^{-1}(I+AD^{-1})^{-1}$. (it's not better using the Woodbury identity).

All the calculations are in $O(n^2)$, except the calculation of $(I+U)^{-1}$ where $U=AD^{-1}$.

or ii) $(A+D)^{-1}=A^*(I+DA^*)^{-1}$. Here all is in $O(n^2)$ except the calculations of $(I+V)^{-1}$, where $V=DA^*$, and of the product of the result by $A^*$.

Then the problem reduces to the calculation of $(I+W)^{-1}$ where $W$ is, roughly speaking, a polar form. Then $W$, a priori, has no particularity. Then the complexity of the previous calculation is $\sim n^3$.

Remark. The hypothesis $||W||<1$ is absolutely useless; to believe the opposite is an urban legend. Indeed $(I+W)^{-1}\approx I-W+W^2-W^3$ has already a complexity $\sim 2n^3$.

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    Its not evident that you have tried to use the fact that $A$ is unitary. Please see my full question, where I have mentioned $A$ is unitary. Hope it will change the answer. – Rajesh D Oct 30 '18 at 14:29