[Solved]: Computing inverse matrix when an element changes

Problem Detail: 

Given an $n \times n$ matrix $\mathbf{A}$. Let the inverse matrix of $\mathbf{A}$ be $\mathbf{A}^{-1}$ (that is, $\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$). Assume that one element in $\mathbf{A}$ is changed (let's say $a _{ij}$ to $a' _{ij}$). The objective is to find $\mathbf{A}^{-1}$ after this change. Is there a method to find this objective that is more efficient than re-calculating the inverse matrix from scratch.

Asked By : AJed

Answered By : Yuval Filmus

The Sherman-Morrison formula could help:

$$ (A + uv^T)^{-1} = A^{-1} - \frac{A^{-1} uv^T A^{-1}}{1 + v^T A^{-1} u}. $$

Let $u = (a'_{ij}-a_{ij}) e_i$ and $v = e_j$, where $e_i$ is the standard basis column vector. You can check that if the updated matrix is $A'$ then $$ A^{\prime -1} = A^{-1} - \frac{(a'_{ij}-a_{ij})A^{-1}_{i\rightarrow} A^{-1T}_{\downarrow j}}{1 + (a'_{ij}-a_{ij})A^{-1}_{ij}}.$$

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Question Source : http://cs.stackexchange.com/questions/9875