I am trying to understand inverses of covariance matrices that are close to each other.
I start with a positive semidefinite matrix $\mathbf{\Sigma^{-1}}$, then I take the pseudo-inverse to get covariance matrix $\mathbf{\Sigma}$. Then this is transform into correlation matrix, $$\mathbf{R = D^{-1}\ \Sigma\ D^{-1}} \ \ \text{where}\ \ \mathbf{D} = \sqrt{\operatorname{diag}(\mathbf{\Sigma})}$$ and then I generate correlated random variables $\mathbf{x}$ which have correlation matrix $\mathbf{R^{'}}$, which is not exact same as $\mathbf{R}$ but is very close. I then convert $\mathbf{R^{'}}$ as above to get $\mathbf{\Sigma^{'}}$ which is close to original $\mathbf{\Sigma}$. I am trying now to invert this $\mathbf{\Sigma^{'}}$ to get the $\mathbf{\Sigma^{'-1}}$ which I would expect is good approximation of $\mathbf{\Sigma}$. However when I actually perform this procedure, I get a $\mathbf{\Sigma^{'}}$ that is indeed close to $\mathbf{\Sigma}$ but the inverse $\mathbf{\Sigma^{'-1}}$ is completely different from $\mathbf{\Sigma^{-1}}$. It is not only the non-zero entries that are different, the pattern of zero is also completely different.
I am doing the above to test out part of an algorithm for coursework assignment, but am not able to proceed because $\mathbf{\Sigma^{'-1}}$ is clearly not even close to $\mathbf{\Sigma^{-1}}$.
I understand from other questions on this site that it is not guaranteed that matrices close to each other will always produce inverses that is close. In this case is there procedure to invert $\mathbf{\Sigma^{'}}$ which will give a good approximation for $\mathbf{\Sigma^{-1}}$?