In the original paper of Conjugate Gradients, the authors mention that if we pick the canonical basis $\{e_1,e_2,\ldots,e_n\}$, to obtain A-orthonormal vectors, we end up with the Gaussian elimination method (section $12$). My question is the following:
Apart from this canonical basis and of-course the Krylov subspace, are there other attractive subspaces to look at?
For instance, intuitively, it looks like if $A$ is well-conditioned, picking the subspace generated by $\{b,A^Tb,(A^T)^2b,(A^T)^3b,\ldots\}$ might be attractive option. If $A$ is the most well-conditioned matrix, the method is guaranteed to converge in $1$ iteration. Has there been some analysis done on other possible subspaces (apart from Krylov and the canonical ones)?
