= Right preconditioning
{title2=$AM^{-1}u=b,\quad x=M^{-1}u$}
<Right preconditioning> substitutes $x=M^{-1}u$ into the <linear system> $Ax=b$. Applying its transformed <linear map> first solves $Mz=v$, then computes $Az$. Recover the solution by solving $Mx=u$. Its <linear-system residual> $b-AM^{-1}u$ is exactly the original <linear-system residual> $b-Ax$, which is useful when monitoring the imbalance in the original equations.
The <left preconditioning> and <right preconditioning> transformed <matrices> are <similar matrices>:
$$
AM^{-1}=M(M^{-1}A)M^{-1}.
$$
They therefore have the same <eigenvalues>, although their <Euclidean norm> <condition numbers> and residual histories need not agree. A <similarity transformation> preserves <eigenvalues>; a nonorthogonal change of coordinates need not preserve lengths.
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