# Can this matrix really be used as a preconditionner?

I've read Boxerman's thesis and I feel that there is possibly a mistake.

We have to resolve $$Ax=b$$ $A$ is a positive-definite symmetric matrix and is very sparse so the conjugate gradient method is chosen.

He wrote (p70 to p73) that to improve the convergence, we can use the following preconditionner : $$P=SC+(I-S)$$ where $C$ is the 3x3 block-diagonal sub-matrix of $A$, $I$ is the identity matrix and $S$ is a 3x3 block-diagonal matrix where each block can be

$S_i=I_3$ or $S_i=0$ or $S_i=I_3-nn^T$ ($n$ is a 3-dimension unit vector)

So, the preconditionner $P$ is a 3x3 Block-Diagonal matrix and can be easily inverted BUT I think it is not necessary symmetric. As far as I know, a preconditionner for the conjugate gradient method must be symmetric.

Am I missing something or there is a mistake in the thesis ?

A simple example could be applying an SPD preconditioner $M$ to solve $Ax=b$ ($A$ SPD) as in the left preconditioning: $M^{-1}Ax=M^{-1}b$. $M^{-1}A$ is clearly neither symmetric nor positive definite and still, it is symmetric with respect to the inner product induced by $M$: $(M^{-1}Ax,y)_M=(MM^{-1}Ax,y)=(Ax,y)=(x,Ay)=(x,MM^{-1}A)=(Mx,M^{-1}Ay)=(x,Ay)_M$. Therefore it is safe to apply CG on $M^{-1}A$ but instead of the standard Euclidean inner product you use the inner product induced by the preconditioner $M$. In fact, this is just what is usually called preconditioned conjugate gradient method (PCG) and is equivalent to applying CG on $M^{-1/2}AM^{-1/2}$.
There are also other exotic preconditioners for CG, e.g., constraint preconditioning particularly popular in optimization. You can apply CG, e.g., on the system in the block form $$A=\begin{bmatrix} K & B \\ B^T & 0 \end{bmatrix},$$ where $K$ is SPD and B has full column rank. The preconditioner $M$ has the same form as $A$ with $K$ replaced by some reasonable approximation $G$, e.g., diagonal. Both the matrix and the precondtitioner are symmetric but indefinite and still CG can be applied since it turns out that it is equivalent to applying CG on $PG^{1/2}KG^{-1/2}P$, where $P$ is a projection onto the range of $B$ (if I remember correctly). You can see more information, e.g., in this article.