Relation between two inverses

Question

Suppose you know $(I+T)^{-1}$, is there any way for approximate the inverse of the matrix $(I+\alpha T)^{-1}$, where $\alpha\in{\mathbb{R}}$?

Answer 1

For $|\alpha-1|\|T(I+T)^{-1}\| < 1$, $$ \begin{align} (I+\alpha T)^{-1} & =((I+T)+(\alpha-1)T)^{-1} \\ & = (I+T)^{-1}(I-(1-\alpha)T(I+T)^{-1}) \\ & = (I+T)^{-1}\sum_{n=0}^{\infty}(1-\alpha)^{n}\{I-(I+T)^{-1}\}^{n} \end{align} $$ That will give you what you want for $\alpha$ in a neighborhood of $1$. In principle, you can continue into other regions where $(I+\alpha T)^{-1}$ exists, but the problems compound with approximation on top of approximation.

Do you know something about the spectrum of $T$, or about bounds on $(I+\alpha T)^{-1}$? That could help.