Jacobi preconditioner not reducing condition number?

Question

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?

Answer 1

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.

If you are using a Krylov method, the global scale is automatically corrected, but with a stationary iteration, the (constant) scaling is needed somehow (could just be in the damping parameter for the stationary method).