Can we really say nothing about the inverse of A+B?
Unfortunately a general formula that simplifies the calculations for a matrix inversion $(A+B)^{-1}$ with arbitrary $A$ and $B$ does not exist. If one of the matrices corresponds to a low-rank update (e.g., $B=CD^T$ for some $C, D\in\mathbb{R}^{n\times k}$ with $k<n$), one can use the S.M.W. formula to great effect. However, in other situations, this formula would not simplify calculations. But all hope is not lost yet. There is another case where $(A+B)^{-1}$ can be computed relatively efficiently. To understand how and…
Demystifying the Sherman-Morrison-Woodbury formula
A matrix inversion formula that frequently appears in machine learning, linear algebra and other textbooks is the Sherman-Morrison-Woodbury formula, according to which a matrix sum can be inverted as \begin{equation}(A+CD^T)^{-1}=A^{-1}-A^{-1}C(I+D^TA^{-1}C)^{-1}D^TA^{-1}.\end{equation}Your first reaction may be wondering if this formula even simplifies anything. The right-hand-side of this equation looks so complicated that one wonders if it’s not simpler to just use compute the sum $A+CD^T$ and invert it. However, this formula can significantly simplify calculations when the matrices $C, D\in \mathbb{R}^{n\times k}$ are…