# what is Sherman-Morrison formula

Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works with Sherman-Morrison formula.

• What do you expect that you cannot trivially find by searching the web? Nov 6, 2018 at 18:07
• Please edit the post with: what do you do not understand? So people may be can help you Nov 7, 2018 at 11:51
• This question is more suited to math.stackexchange, there in the linear-algebra and numerical mathematics section.
– Nox
Nov 7, 2018 at 13:25

Sherman-Morrison formula helps to find the inverse of the matrix cheap after a rank-1 update. Sherman-Morrison-Woodbury does it for a low-rank update (not necessarily rank-1).

In short,

1. You have a matrix $$A$$ and you already have a computed $$A^{-1}$$.
2. For some reason, you need to do a rank-1 update to $$A$$. This rank-1 update can be described as an outer-product of two vectors $$u$$ and $$v$$.
3. So, you want an inverse of $$(A+uv^T)$$, but you don't want to redo all the calculations. Can you reuse your knowledge of $$A^{-1}$$? Yes.
4. Sherman-Morrison tells you that $$(A+uv^T)^{-1} = A^{-1}-\frac{A^{-1}uv^TA^{-1}}{1+v^tA^{-1}u}$$ which allows huge savings since you are just performing relatively cheap manipulations with an already obtained $$A^{-1}$$.

I strongly encourage you to read at least the Wikipedia articles regarding this topic in order to get an understanding about

• numerical stability
• requirements for the formula to be applicable

Important notes:

• calculations of the inverse should be avoided at all costs
• same ideas (SM, SMW) usually can be applied to factorizations, where they are extremely useful