Condition number of preconditioned system Suppose we are solving an ill-conditioned system $Ax = b$, and we are trying to solve it using preconditioned technique. Given $\kappa (T)\approx \kappa(A)$, where $\kappa(A)$ is condition number of $A$ w.r.t some matrix norm. Can we show that $\kappa((T+\Delta T)^{-1}A)\ll \kappa(A)$, where $|\Delta T|\leq \delta |T|$, $\delta\ll 1$?

You need more information to eliminate pathological cases such as $T = \text{diag}(1,1,1,....,1,\epsilon)$ and $A = \text{diag}(1,1,1,...,\epsilon^{-1})$. Both matrices have condition number $\epsilon^{-1}$, but \begin{equation} T^{-1}A = \text{diag}(1,1,1,...,1,\epsilon^{-2}) \end{equation} has condition number $\epsilon^{-2}$.