I'm hoping that someone can point me to some literature on the following.
Is there a way to bound the inverse of a matrix if I change the value of 1 element in that matrix. Let's say I have a matrix $A$ and its inverse $A^{-1}$. If I replace element $A(1,1)$ with $A(1,1)+\delta$ (and assuming the inverse still exists), is there some way to relate the error in the inverse of my perturbed matrix to the true inverse?
thanks
If you replace $A$ by $A+δ e_1e_1^T$, then by the formulas of Sherman, Woodbury and Morrison the determinant obeys $\det(I+uv^T)=1+v^Tu$, i.e.,
$$\det(A+δ e_1e_1^T)=\det(A)(1+δe_1^TA^{-1}e_1)$$
and the inverse follows from $(I+uv^T)(I+cuv^T)=I+(1+c+cv^Tu)uv^T$ for $c=-(1+v^Tu)^{-1}$, that is for the perturbed $A$
\begin{align} (A+δ\, e_1e_1^T)^{-1} &=A^{-1}(I-(1+δ\,e_1^TA^{-1}e_1)^{-1}\,δ\,e_1e_1^TA^{-1})\\ &=\frac{A^{-1}(1+δ\,e_1^TA^{-1}e_1)-δ\,A^{-1}e_1e_1^TA^{-1}}{1+δ\,e_1^TA^{-1}e_1} \end{align}
So a very coarse estimate is
$$\|(A+δ\, e_1e_1^T)^{-1}\|\le\frac{\|A^{-1}\|\,(1+2δ\|A^{-1}\|)}{1-δ\|A^{-1}\|}$$
Using the first form of the inverse, this can be improved to \begin{align} \|(A+δ\, e_1e_1^T)^{-1}\| &\le\|A^{-1}\|\left(1+\frac{δ\|A^{-1}\|}{1-δ\|A^{-1}\|}\right)\\ &=\frac{\|A^{-1}\|}{1-δ\|A^{-1}\|} \end{align}