 Matrix preconditioning

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

A useful matrix preconditioner combines a cheap solve with favorable transformed eigenvalues; neither property alone is enough.^[ref]

The aim is to make each correction more effective without making it expensive. The example under diagonal scaling for a two-dimensional linear system turns thin ellipses into nearly circular ones and makes preconditioned steepest descent effective.