A note on preconditioning for indefinite linear systems

Preconditioners are often conceived as approximate inverses. For nonsingula r indefinite matrices of saddle-point (or KKT) form, we show how preconditi oners incorporating an exact Schur complement lead to preconditioned matric es with exactly two or exactly three distinct eigenvalues. Thus approximati ons of the Schur complement lead to preconditioners which can be very effec tive even though they are in no sense approximate inverses.