Matrix preconditioner ($M$)

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 $M^{-1/2}A{M}^{-1/2}$.

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.^[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.^[ref]

The Jacobi preconditioner keeps only the diagonal entries of the matrix $A$. For a symmetric positive-definite matrix, these entries are positive, so its auxiliary solve is $z_i=r_i/a_{ii}$. It rescales coordinates with very different diagonal sizes, but cannot remove arbitrary off-diagonal coupling.^[ref]

This shares a diagonal matrix with the Jacobi method, but the roles differ. The Jacobi method specifies a whole stationary iterative method for a linear system; a Jacobi preconditioner supplies an auxiliary solve to another iterative method, including the Preconditioned conjugate gradient method.