<s>[INST] Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$.  Let $D$ be a diagonal matrix with the same dimensions and rank.  What is the fastest way to compute $(A+D)^{-1}$?  Assume all diagonal elements of $A+D$ are positive and that $A+D$ is invertible.
 [/INST] Since $(A+D)^{-1} = (I+ A^{-1} D)^{-1} A^{-1}$, and $A^{-1}$ is known, we just need to find out how to calculate $B ^{-1}$ where $B$ is the (known) matrix $1+A^{-1} D$.
Following Greg Martin's observation, consider first the case where the only nonzero entry of $D$ is in the upper left corner. In this case, all columns of $A^{-1} D$ are zero except the first one so $B$ has the form
$$ B = \begin{pmatrix} 
v_1 & 0 & 0 &\cdots & 0\\
v_2 & 1 & 0 & \cdots & 0 \\
v_3 & 0 & 1 & \cdots & 0 \\
\vdots & & & & \vdots \\
v_n & 0 & 0 &  \cdots & 1 \\
\end{pmatrix} = 
\begin{pmatrix} 
v_1 & 0 & 0 &\cdots & 0\\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & & & & \vdots \\
0 & 0 & 0 &  \cdots & 1 \\
\end{pmatrix} 
\begin{pmatrix} 
1 & 0 & 0 &\cdots & 0\\
v_2 & 1 & 0 & \cdots & 0 \\
v_3 & 0 & 1 & \cdots & 0 \\
\vdots & & & & \vdots \\
v_n & 0 & 0 &  \cdots & 1 \\
\end{pmatrix}$$ 
Invertibility of $B$ is equivalent to the condition $v_1 \neq 0$. In this case both factors above are invertible. The inverse of the diagonal factor on the left is trivial to compute, and the inverse of the right factor happens to be
$$ \begin{pmatrix} 
1 & 0 & 0 &\cdots & 0\\
-v_2 & 1 & 0 & \cdots & 0 \\
-v_3 & 0 & 1 & \cdots & 0 \\
\vdots & & & & \vdots \\
-v_n & 0 & 0 &  \cdots & 1 \\
\end{pmatrix}$$
so the inverse of $B$ is easy to recover. Namely:
$$B^{-1} = \begin{pmatrix} 
\frac{1}{v_1} & 0 & 0 &\cdots & 0\\
-\frac{v_2}{v_1} & 1 & 0 & \cdots & 0 \\
-\frac{v_3}{v_1} & 0 & 1 & \cdots & 0 \\
\vdots & & & & \vdots \\
-\frac{v_n}{v_1} & 0 & 0 &  \cdots & 1 \\
\end{pmatrix}.$$
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