I posted the following question on stackexchange but didn't get any replies; I'm hoping perhaps someone can help me here.


I understand that for many iterative methods, convergence rates can be shown to depend on the condition number of the coefficient matrix $A$ in the linear equation 
$$Ax=y.$$
Therefore, if a preconditioner satisfies $P \approx A$, then by solving the transformed linear equation
$$(AP^{-1}) (Px)=y.$$
the new coefficient matrix will now have more favorable spectral properties and hence better convergence can be achieved. 


One of the main properties a good preconditioner should satisfy besides the above condition is that its inverse should be cheap to apply. Thus, they are often sought out for with a certain structure. Typical examples are the incomplete Cholesky and LU factorizations of the matrix $A$.


My question is: why do we want to have $P \approx A$, or, in a more direct approach, why do we formulate finding preconditioners as:
$$
\min\_{P} \left\| AP^{-1} - I \right\|\_F,
$$
where $F$ represents the Frobenius-norm? The identity matrix isn't the only one with a condition number of 1; would it not be better to formulate the problem as:
$$
\min\_{P,Q} \left\| AP^{-1} - Q \right\|\_F,
$$
with $Q$ having to be orthogonal? Given a certain structure restriction on $P$, I imagine this would lead to better preconditioning than in the previous case.



Question about preconditioning
The Frobenius norm remains the same if we do the orthogonal transform, so 
$$
\min\_{P,Q} \left\| AP^{-1} - Q \right\|\_F =
\min\_{P,Q} \left\| Q^{-1}AP^{-1} - I \right\|\_F
$$
which is essentially nothing more than a two-sided preconditioning. While for some reason you consider only right-sided in the beginning (usually I see the left preconditioning but it depends on the problem, of course).


