When our system of equations
\( Ax=b \)
is difficult to solve, then we can construct a preconditioner \( M \) (another matrix) such that if we multiply our equation on the left by \( M^{-1} \) czyli
\( M^{-1}Ax=M^{-1}b \)
and additionally in between \( Ax \)
we put \( M^{-1}M \) (we can do it because \( M^{-1}M \) is an identity matrix), then a new system of equations
\( M^{-1} A M^{-1} M x = M^{-1} b \)
may be easier to solve. Of course, it all depends on the clever structure of the matrix \( M \).
An example is given in the Iterative solver chapter. Preconditioners for iterative algorithms are described in chapter 8 of the textbook by Yousef Saad [1].