I am working on an optimization problem (for example, conjugate gradient) to solve $Ax=b$, where $A$ is a symmetric positive definite matrix. I can understand that the CG (conjugate gradient) has better performance when the matrix $A$ has a smaller conditioner number. But I am wondering is there a relationship between the eigenvectors corresponding to the largest few eigenvalues and the first few update directions of the CG? Any suggestions would be helpful. Thanks!

(This question is also posted here.)