Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 40E / d / Solution

Because $A$ is a positive-definite matrix, it has a positive-definite inverse square root. Choose the exact inverse-square-root preconditionerThenwhich has one distinct eigenvalue, so preconditioned CG terminates in one step.

This construction is of little computational use because obtaining and applying $A^{-1/2}$ is generally at least as expensive in storage and arithmetic as directly factoring or solving the original system. A useful preconditioner must only approximate this transformation while remaining cheap to construct and apply.