Matrix preconditioning ($M^{-1/2}A{M}^{-1/2}$)

Matrix preconditioning transforms a system using an inexpensive auxiliary solve to improve its spectrum. For symmetric positive definite matrices, a positive definite $M$ produces $M^{-1/2}A{M}^{-1/2}$, preserving the symmetric structure needed by Conjugate gradient method.

A matrix preconditioner is an auxiliary invertible matrix whose inverse is cheap to apply and whose use improves the spectrum of a linear system. In symmetric positive definite problems, a positive definite $M$ preserves that structure under the symmetric transformation $M^{-1/2}A{M}^{-1/2}$.