= Matrix preconditioning
{title2=$M^{-1/2}AM^{-1/2}$}
<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 $M^{-1/2}AM^{-1/2}$, 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.<Math 221, Matrix Computations, Fall 2023/Notes for Ma221 Lecture 15, Dec 1, 2023>[Math 221, Matrix Computations, Fall 2023; Notes for Ma221 Lecture 15, Dec 1, 2023; PDF p. 1, opening transformation, goals (1) and (2), and example (1)]{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.
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