= Left preconditioning
{title2=$M^{-1}Ax=M^{-1}b$}
<Left preconditioning> multiplies both sides of the <linear system> $Ax=b$ by the same <invertible matrix> $M^{-1}$. It leaves the unknown <vector> $x$ unchanged. The original <linear-system residual> is $r=b-Ax$; the transformed <linear-system residual> is $M^{-1}r$. Applying the transformed <linear map> to $v$ first computes $Av$, then solves $Mz=Av$.
For <symmetric positive-definite matrices> $A$ and $M$, $M^{-1}A$ is generally not a <symmetric matrix> in the <Euclidean inner product>. Its positive <eigenvalues> alone do not justify ordinary Euclidean <Conjugate gradient method> recurrences. The remedy is <symmetric positive-definite preconditioning>, implemented through the <Preconditioned conjugate gradient method>.
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