= Matrix preconditioner
{title2=$M$}
A <matrix preconditioner> is an auxiliary <invertible matrix> whose <matrix inverse> is cheap to apply and whose use improves the <eigenvalues> of a <linear system>. In <symmetric positive-definite matrix> problems, a <symmetric positive-definite matrix> $M$ preserves that structure under the symmetric transformation $M^{-1/2}AM^{-1/2}$.
Applying $M^{-1}$ means solving $Mz=r$ for a supplied <vector> $r$. This notation does not require forming a general <matrix inverse>. A <diagonal matrix> gives componentwise divisions; a factored <matrix> gives auxiliary solves such as <forward substitution> and <back substitution>.<Math 221, Matrix Computations, Fall 2009/Notes for Ma221 Lecture 21, Nov 10 2009>[Math 221, Matrix Computations, Fall 2009; Basic iterative methods, web lines 297–308, recurrence (\*) and cheap auxiliary-solve requirement]{ref}
Choosing $M=I$ makes the solve trivial but changes nothing; choosing $M=A$ gives the exact correction but asks us to solve the original <linear system>.<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, contrasting M = I and M = A; p. 2, example (3), two triangular solves]{ref}
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