# Jacobi preconditioner not reducing condition number?

Let's say you have a general matrix $$A$$, with diagonal entries $$a_{ii} = d>0$$. (No assumptions are made about the off-diagonal elements.) Then Jacobi preconditioning doesn't improve condition number, because $$\kappa(A)= \left|\frac{\lambda_{\rm{max}}(A)}{\lambda_{\rm{min}}(A)} \right|$$ becomes $$\kappa(M^{-1}A)= \left|\frac{\lambda_{\rm{max}}(A)/d}{\lambda_{\rm{min}}(A)/d} \right| = \kappa(A).$$ In this case, could the use of a Jacobi preconditioner do anything useful?

No, Jacobi only ever corrects relative scales. It does nothing for "smooth" ill-conditioning, such as the $\kappa(A) \in O(h^{-2})$ asymptotics for second order elliptic problems.