Question about preconditioning 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.
 A: 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).
A: The answer is that looking for a matrix such that $AP^{-1} \approx I$ is a simplification. The convergence speed of Krylov subspace methods depends heavily on the clustering of eigenvalues of $AP^{-1}$ (in the normal case at least; in the non-normal case it's not that clear but clustering eigenvalues seems to be a good proxy).
If you ensure that $\|AP^{-1}-I\|$ is small, then you also automatically ensure that the eigenvalues are clustered around 1. But that's not the only possibility.
Some preconditioning strategies for saddle-point problems, for instance, aim to obtain a matrix with three clusters of eigenvalues around $1, \frac{1+\sqrt5}2,\frac{1-\sqrt5}2$, and some others aim to obtain two clusters around $-1$ and $1$.
On the other hand, $AP^{-1}$ being orthogonal does not ensure faster convergence. The classical counterexample is the system (after preconditioning)
$$
\begin{bmatrix}
0 & && & 1\\
1 & 0 & \\
 & 1 & 0 &  \\
& & \ddots & \ddots \\
& & & 1 & 0
\end{bmatrix}x = 
\begin{bmatrix}
1\\0\\0\\\vdots\\0
\end{bmatrix},
$$
in which Arnoldi methods stagnate for $n-1$ steps and then converge abruptly in the last one.
